Theory and Applications of Categories, Vol. 32, No. 28, 2017, pp. 970–994.

A URYSOHN TYPE LEMMA FOR GROUPOIDS

MĂDĂLINA ROXANA BUNECI

Abstract. Starting from the observation that through groupoids we can express in a unified way the notions of fundamental system of entourages of a uniform structure on a space X, respectively the system of neighborhoods of the unity of a topological group that determines its topology, we introduce in this paper a notion of G-uniformity for a groupoid G. The topology induced by a G-uniformity turns G into a topological locally transitive groupoid.

We also prove a Urysohn type lemma for groupoids and obtain metrization theorems for groupoids unifying in two ways the Alexandroff–Urysohn Theorem and Birkhoff-Kakutani Theorem.

1. Introduction and preliminaries

The notion of groupoid is a natural generalization of the notion of group in the following sense: a groupoid is a set G endowed with partially defined product operation (x, y) ↦ xy [: G^{(2)} → G] (where G^{(2)} ⊂ G × G) and an inversion operation x ↦ x^{−1} [: G → G] satisfying the subsequent weaker versions of the group axioms:

G1 If (x, y) ∈ G^{(2)} and (y, z) ∈ G^{(2)}, then (xy, z) ∈ G^{(2)}, (x, yz) ∈ G^{(2)} and (xy) z = x (yz).

G2 (x^{−1})^{−1} = x for all x ∈ G.

G3 For all x ∈ G, (x, x^{−1}) ∈ G^{(2)}, and if (z, x) ∈ G^{(2)}, then (zx) x^{−1} = z.

G4 For all x ∈ G, (x^{−1}, x) ∈ G^{(2)}, and if (x, y) ∈ G^{(2)}, then x^{−1} (xy) = y.

The maps r and d on G, defined by the formulae r (x) = xx^{−1} and d (x) = x^{−1}x, are called the range (target) map, respectively the domain (source) map. They have a common image called the unit space of G and denoted G^{(0)}. The fibres of the range and the domain maps are denoted G^{u} = r^{−1} ({u}) and G_{v} = d^{−1} ({v}), respectively. Also for u, v ∈ G^{(0)}, G_{v}^{u} = G^{u} ∩ G_{v}.

A topological groupoid is a groupoid G together with a topology on G such that the product operation (x, y) ↦ xy [: G^{(2)} → G] (where G^{(2)} ⊂ G × G is endowed with the topology induced by the product topology on G × G) and the inversion operation x ↦ x^{−1} [: G → G] are continuous functions. A family {W_{j}}_{j∈J} of neighborhoods of the unit space is said to be compatible with the topology of the r-fibres (respectively, d-fibres) if for every u ∈ G^{(0)} and every open neighborhood U of u, there is j ∈ J such that W_{j} ∩ G^{u} ⊂ U ∩ G^{u} and u is in the interior of W_{j} ∩ G^{u} with respect to the topology on G^{u} coming from G (respectively, W_{j} ∩ G_{u} ⊂ U ∩ G_{u} and u is in the interior of W_{j} ∩ G_{u} with respect to the topology on G_{u} coming from G).

Received by the editors 2016-11-03 and, in final form, 2017-07-31.

Transmitted by Ronald Brown. Published on 2017-08-02.

2010 Mathematics Subject Classification: 22A22; 54E15, 54E35.

Key words and phrases: groupoid, Urysohn-type lemma, metrization theorem.

\begin{matrix}c \\ ◯ \end{matrix} Mădălina Roxana Buneci, 2017. Permission to copy for private use granted.

Let us also recall that a uniform space is a set X endowed with a uniform structure. A fundamental system of symmetric entourages of a uniform structure on X is a nonempty family W of subsets of the Cartesian product X ×X that satisfies the following conditions:

U1 if W is in W, then W contains the diagonal Δ = {(x, x) : x ∈ X}.

U2 if W_{1} and W_{2} are in W, then there is W_{3} ∈ W such that W_{3} ⊂ W_{1} ∩ W_{2}.

U3 if W_{1} is in W, then there exists W_{2} in W such that, whenever (x, y) and (y, z) are in W_{2}, then (x, z) ∈ W_{1}.

U4 if W ∈ W, then W = W^{−1} = {(y, x) : (x, y) ∈ W } (W is a symmetric entourage).

The uniform space X becomes a topological space by defining a subset A ⊂ X to be open if and only if for every x ∈ A there is W_{x} ∈ W such that {y : (x, y) ∈ W_{x}} ⊂ A.

The Cartesian product X × X can be viewed as a trivial groupoid G under the operations: (x, y) (y, z) = (x, z) and (x, y)^{−1}. In the settings of groupoids condition U1 can be written as ”G^{(0)} ⊂ W ⊂ G for all W ∈ W” and condition U3 as ”for every W_{1} ∈ W there is W_{2} ∈ W such that W_{2}W_{2} ⊂ W_{1}”.

In this paper we work with a collection of subsets of a groupoid G mimicking the properties of fundamental system of symmetric entourages of a uniform structure on X. Such a collection will be called in this paper G-uniformity. We prove that a G-uniformity induces a topology on G that turns G into a topological locally transitive groupoid. Let us recall that a topological locally transitive groupoid is a topological groupoid G with the property that for all u ∈ G^{(0)} the maps r_{u} are open, where r_{u} : G_{u} → G^{(0)}, r_{u} (x) = r (x) for all x ∈ G_{u} and G_{u} is endowed with the topology coming from G (see [12]). If we begin with a topological groupoid (G, τ) and with a G-uniformity given by a fundamental system of neighborhoods of the unit space, then the topology induced by de G-uniformity is finer than τ and coincides with τ if and only if (G, τ) is locally transitive. The main result of this paper is a Urysohn type lemma for groupoids (Theorem 2.5). The existence of a function with properties 1 − 3 in Theorem 2.5 could also be obtained taking into account that a G-uniformity is a base for a uniform structure on G. However the topology defined by the G-uniformity do not necessarily coincides with the groupoid topology, even if the G-uniformity is given by a fundamental system of neighborhoods of the unit space. The construction in Theorem 2.5 allows us to get a function with additional properties. In particular, in the case of a topological groupoid with open range map and a G-uniformity given by a fundamental system of neighborhoods of the unit space, our construction allows us to put out a connection with the groupoid topology: the functions f associated in Theorem 2.5 with open subsets of G or with G^{(0)} are upper semi-continuous on G and their restrictions to the r-fibres as well as to the d-fibres of the groupoid are continuous functions. Thus these functions can be used to construct convolutions algebras as in [4] and possibly to extend the construction of a C^{∗}-algebra associated to a topological locally compact groupoid with continuous Haar system introduced in [11]. Moreover the property 9 in Theorem 2.5 allows us to obtain metrization theorems for groupoids and thus to express in an unified way Alexandroff–Urysohn Theorem and Birkhoff-Kakutani Theorem as we explain below. Let us consider the following two theorems:

1.1. Theorem. [Alexandroff–Urysohn Theorem] A topological Hausdorff space X is metrizable if and only if its topology is given by a uniformity with countable base. [1]

1.2. Theorem. [Birkhoff-Kakutani Theorem] A topological group G is metrizable if and only if there is a countable base for the topology at identity element in G. Furthermore, in such a case, the distance function may be taken to be either left-invariant or rightinvariant. ([2], [6])

Let us remark that the space X, respectively the group G can be viewed as r -fibres (as well as d-fibres) of a groupoid (X ×X in the first case and G itself in the second case). We prove in this paper that the previous two results can be express in an unified way in the groupoid language:

1.3. Theorem. Let G be a topological groupoid. Then there are left (respectively, right) invariant metrics compatible with the topology on r-fibres (respectively, the d-fibres) of the groupoid if and only if there is a countable G-uniformity {W_{n}}_{n∈ℕ} compatible with the topology of the r-fibres (respectively, d-fibres) such that ⋂_{n∈ℕ} W_{n} = G^{(0)}. (Proposition 3.14

and Proposition 3.15)

The proof of this theorem is based on the construction of a function on G satisfying the hypothesis of [8, Theorem 3.26]. This function is obtained as a particular case of Urysohn Lemma for groupoids (Theorem 2.5).

We also prove in this paper that:

1.4. Theorem. For a topological locally transitive groupoid G the following statements are equivalent:

(a) G is metrizable

(b) For every neighborhood W of G^{(0)} there is a neighborhood W′ of G^{(0)} such that W′W′ ⊂ W and G^{(0)} has a countable fundamental system {W_{n}}_{n∈ℕ} of neighborhoods such that

⋂_{n∈ℕ} W_{n} = G^{(0)} and ⋂_{n∈ℕ} (r, d) (W_{n}) = diag (G^{(0)}).

(c) There is a countable G-uniformity {W_{n}}_{n∈ℕ} compatible with the topology of the fibres such that ⋂_{n∈ℕ} W_{n} = G^{(0)} and ⋂_{n∈ℕ} (r, d) (W_{n}) = diag (G^{(0)}). Each W_{n} may be taken

to be a neighborhood of the unit space.

A URYSOHN TYPE LEMMA FOR GROUPOIDS  973

Moreover the distance function ρ may be taken to satisfy the following properties:

1. ρ (x, y) = ρ (x^{−1}, y^{−1}) for all x, y ∈ G.

2. ρ (x, r (x)) = ρ (x, d (x)) for all x ∈ G.

3. ρ (xy, r (x)) ≤ ρ (x, r (x)) + ρ (y, r (y)) for all (x, y) ∈ G^{(2)}.

4. ρ (x, y) ≤ ρ (x^{−1}y, d (x)) for all x, y ∈ G such that r (x) = r (y).

5. ρ (d (x) , d (y)) ≤ 2ρ (x, y) and ρ (r (x) , r (y)) ≤ 2ρ (x, y) for all x,y ∈ G. (Theorem 3.16)

2. Urysohn’s lemma for groupoids

2.1. Definition. Let G be a groupoid. By a G-uniformity we mean a collection {W }_{W ∈W} of subsets of G satisfying the following conditions:

1. G^{(0)} ⊂ W ⊂ G for all W ∈ W.

2. If W_{1}, W_{2} ∈ W, then there is W_{3} ⊂ W_{1} ∩ W_{2} such that W_{3} ∈ W.

3. For every W_{1} ∈ W there is W_{2} ∈ W such that W_{2}W_{2} ⊂ W_{1}.

4. W = W^{−1}for all W ∈ W. 2.2. Definition. Let G be a groupoid. Two G-uniformities W and W′ are said to be equivalent if for every W ∈ W there is W′ ∈ W′ such that W′ ⊂ W and conversely, for every W′ ∈ W′ there is W ∈ W such that W ⊂ W′.

Let W be a family of subsets of a groupoid G satisfying conditions 1−4 from Definition 2.1 and let

I = \{\frac{1}{2^{n}} , n ∈ ℕ\}

Let W_{0} ∈ W and W_{1} ∈ W be such that W_{1}W_{1} ⊂ W_{0}. Inductively we can construct an I-indexed family {W_{i}}_{i∈I}. Suppose that for W_{i} ∈ W has already been built. Then according condition 3 in Definition 2.1, there is a W′_{i} ∈ W such that W′_{i}W′_{i} ⊂ W_{i}. Let W_{i/2} = W′_{i}. Thus we obtain an I-indexed family {W_{i}}_{i∈I} satisfying the following properties:

1. W_{i} ∈ W for all i ∈ I.

2. W_{i}W_{i} ⊂ W_{2i} for all i ∈ I, i ≤ \frac{1}{2}.

3. W_{1}W_{1} ⊂ W_{0}.

Hence W_{i} ⊂ W_{i}W_{i} ⊂ W_{2i} for all i ∈ I, i ≤ \frac{1}{2} and ...W_{1/2^{n}} ⊂ W_{1/2^{n}}W_{1/2^{n}} ⊂ W_{1/2^{n−1}} ⊂ W_{1/2^{n−1}}W_{1/2^{n−1}} ⊂ ...W_{1/2} ⊂ W_{1/2}W_{1/2} ⊂ W_{1} Let us note that:

1. If i, j ∈ I, then i < j iff there is p ∈ ℕ^{∗} such that j = 2^{p}i.

2. If i, j ∈ I and i < j, then 2i ≤ j.

3. If i, j ∈ I and i ≤ j, then W_{i} ⊂ W_{j}.

4. If i_{1}, i_{2}, ..., i_{k} ∈ I and i_{k} ≤ i_{k−1} < i_{k−2} < ... < i_{1} < 1, then W_{i_{k}}W_{i_{k−1}}...W_{i_{1}} ⊂ W_{2i_{1}} and W_{i_{1}}...W_{i_{k−1}}W_{i_{k}} ⊂ W_{2i_{1}}. Indeed,

W_{i_{k}}W_{i_{k−1}}W_{i_{k−2}}...W_{i_{1}} ⊂ W_{i_{k−1}}W_{i_{k−1}}W_{i_{k−2}}...W_{i_{1}}
⊂ W_{2i_{k−1}}W_{i_{k−2}}...W_{i_{1}}
⊂ W_{i_{k−2}}W_{i_{k−2}}...W_{i_{1}}
⊂ ... ⊂ W_{2i_{1}}

Similarly, W_{i_{1}}...W_{i_{k−1}}W_{i_{k}} ⊂ W_{i_{1}}...W_{i_{k−1}}W_{i_{k−1}} ⊂ W_{i_{1}}...W_{i_{k−2}}W_{2i_{k−1}} ⊂ ...W_{2i_{1}}.

5. If i_{1}, i_{2}, ..., i_{k}, j_{1}, j_{2}, ..., j_{m} ∈ I, i_{k} < i_{k−1} < ... < i_{1} ≤ 1, j_{m} < j_{m−1} < ... < j_{1} ≤ 1 and i_{k} + i_{k−1} + ... + i_{1} ≤ j_{m} + j_{m−1} + ... + j_{1}, then

W_{i_{k}}W_{i_{k−1}}...W_{i_{1}} ⊂ W_{j_{m}}W_{j_{m−1}}...W_{j_{1}}.

Indeed, let us remark that i_{k} +i_{k−1} +...+i_{1} = \frac{1}{2^{n_{k}}} + \frac{1}{2^{n_{k−1}}} +...+ \frac{1}{2^{n_{1}}} is the conversion into decimal system of the following number in base 2: b_{0}, b_{1}b_{2}...b_{n_{k}} where b_{i} = 1 if i ∈ {n_{1}, n_{2}, ..., n_{k}} and b_{i} = 0 otherwise. Thus if i_{k}+i_{k−1}+...+i_{1} = j_{m}+j_{m−1}+...+j_{1}, then m = k and i_{k} = j_{k}, ..., i_{1} = j_{1}. If i_{k} + i_{k−1} + ... + i_{1} < j_{m} + j_{m−1} + ... + j_{1}, then there is p ∈ ℕ^{∗} such that i_{1} = j_{1}, ..., i_{p−1} = j_{p−1} and i_{p} < j_{p}. Hence

W_{i_{k}}W_{i_{k−1}}...W_{i_{p}}W_{i_{p−1}}... W_{i_{1}} ⊂ W_{2i_{p}}W_{i_{p−1}}... W_{i_{1}}
⊂ W_{j_{p}}W_{i_{p−1}}... W_{i_{1}}
= W_{j_{p}}W_{j_{p−1}}... W_{j_{1}}
⊂ W_{j_{m}}W_{j_{m−1}}...W_{j_{1}}.

2.3. Lemma. Let G be a groupoid, W be a G-uniformity (in the sense of Definition 2.1) and let

I = \{\frac{1}{2^{n}} , n ∈ ℕ\}.

Let us consider an I-indexed family {W_{i}}_{i∈I} satisfying the following properties:

1. W_{i} ∈ W for all i ∈ I.

2. W_{i}W_{i} ⊂ W_{2i} for all i ∈ I, i ≤ \frac{1}{2}.

For W_{i_{k}}, W_{i_{k−1}}, ..., W_{i_{1}} ∈ {W_{i}}_{i∈I}, let us denote

s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) = i_{k} + i_{k−1} + ... + i_{1}.

Let n ∈ ℕ^{∗}, and i_{1,} i_{2}, ..., i_{k} ∈ I be such that i_{k} < i_{k−1} < i_{k−2} < ... < i_{1} < 1. Then there are j_{1}, j_{2}, ..., j_{r} ∈ I such that

1. j_{r} < j_{r−1} < i_{r−2} < ... < j_{1} ≤ max \{\frac{1}{2^{n−1}} , 2i_{1}\} ≤ 1

2. W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{1}} ⊂ W_{j_{r}}W_{j_{r−1}}...W_{j_{1}}

3. 0 < s (W_{j_{r}}W_{j_{r−1}}...W_{j_{1}}) − s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) ≤ \frac{1}{2^{n−1}}

Moreover j_{1} < s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n−1}} and if j_{r} ≠ \frac{1}{2^{n}}, then j_{r} ≥ \frac{1}{2^{n−1}} . Also if \frac{1}{2^{n}} ≤ i_{k}, then s (W_{j_{r}}W_{j_{r−1}}...W_{j_{1}}) − s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) ≤ \frac{1}{2^{n}}.

Proof. Case 1: \frac{1}{2^{n}} < i_{k}. Obviously, \frac{1}{2^{n}} < i_{k} < i_{k−1} < i_{k−2} < ... < i_{1} < 1 and s (W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) = s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n}}.

Case 2: There is m ∈ \{2, 3, ..., k\} such that i_{m} = \frac{1}{2^{n}} < \frac{i_{m−1}}{2} . Then

W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}} ⊂ W_{1/2^{n}}W_{2i_{m}}W_{i_{m−1}}...W_{i_{1}}.

and we have

s (W_{1/2^{n}}W_{2i_{m}}W_{i_{m−1}}...W_{i_{1}}) =

= s (W_{i_{k}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}}) − (i_{k} + ... + i_{m}) + 2i_{m} + \frac{1}{2^{n}}
≤ s (W_{i_{k}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}}) + i_{m} + \frac{1}{2^{n}}
= s (W_{i_{k}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}}) + \frac{2}{2^{n}}.

Moreover i_{k} + ... + i_{m} ≤ (\frac{1}{2^{k−m}} + \frac{1}{2^{k−m}} + ...\frac{1}{2} + 1) i_{m} < 2i_{m} < 2i_{m} + \frac{1}{2^{n}}. Consequently, s (W_{1/2^{n}}W_{2i_{m}}W_{i_{m−1}}...W_{i_{1}}) > s (W_{i_{k}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}}).

Case 3: There is m ∈ \{2, 3, ..., k\} such that i_{m} = \frac{1}{2^{n}} = \frac{i_{m−1}}{2} and there is q ∈ {2, 3, ..., m − 1} such that 4i_{q} ≤ i_{q−1}. Let p be the greatest element of the set

{q : 2 ≤ q ≤ m − 1, 4i_{q} ≤ i_{q−1}}.

Then W_{1/2^{n}}W_{i_{k}}...W_{i_{1}} ⊂ W_{1/2^{n}}W_{2i_{m}}W_{i_{m−1}}...W_{i_{1}} ⊂ W_{1/2^{n}}W_{2i_{p}}W_{i_{p−1}}...W_{i_{1}}. Moreover

s (W_{2i_{p}}W_{i_{p−1}}...W_{i_{1}}) = s (W_{i_{m−1}}...W_{i_{p}}W_{i_{p−1}}...W_{i_{1}}) − (i_{m−1} + .. + i_{p}) + 2i_{p}

= s (W_{i_{m−1}}...W_{i_{p}}W_{i_{p−1}}...W_{i_{1}}) − (i_{m−1} + 2i_{m−1} + ... + 2^{m−p−1}i_{m−1}) + 2^{m−p}i_{m−1}
= s (W_{i_{m−1}}...W_{i_{p}}W_{i_{p−1}}...W_{i_{1}}) − i_{m−1} (2^{m−p} − 1) + i_{m−1}2^{m−p}
= s (W_{i_{k}}...W_{i_{p}}W_{i_{p−1}}...W_{i_{1}}) − (i_{k} + ... + i_{m}) + i_{m−1},

and since \frac{1}{2^{n−1}} = i_{m−1}, it follows that

s (W_{1/2^{n}}W_{2i_{p}}W_{i_{p−1}}...W_{i_{1}}) = s (W_{2i_{p}}W_{i_{p−1}}...W_{i_{1}}) + \frac{1}{2^{n}}

= s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) − (i_{k} + ... + i_{m}) + i_{m−1} + \frac{1}{2^{n}}
= s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) − (i_{k} + ... + i_{m+1}) − \frac{1}{2^{n}} + \frac{1}{2^{n−1}} + \frac{1}{2^{n}}
= s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) − (i_{k} + ... + i_{m+1}) + \frac{1}{2^{n−1}} .

On the other hand, i_{k} + ... + i_{m} ≤ (\frac{1}{2^{k−m+1}} + \frac{1}{2^{k−m}} + ...\frac{1}{2}) i_{m−1} < i_{m−1} and therefore s (W_{1/2^{n}}W_{2i_{p}}W_{i_{p−1}}...W_{i_{1}}) > s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}).

Case 4: There is m ∈ \{2, 3, ..., k\} such that i_{m} = \frac{1}{2^{n}} = \frac{i_{m−1}}{2} = \frac{i_{m−2}}{2^{2}} = ... = \frac{i_{1}}{2^{m−1}} . Then W_{1/2^{n}}W_{i_{k}}...W_{i_{1}} ⊂ W_{1/2^{n}}W_{2i_{m}}W_{i_{m−1}}...W_{i_{1}} ⊂ W_{1/2^{n}}W_{2i_{1}} and

s (W_{1/2^{n}}W_{2i_{1}}) = s (W_{i_{m}}...W_{i_{1}}) − (i_{m} + .. + i_{1}) + 2i_{1} + \frac{1}{2^{n}}
= s (W_{i_{m}}...W_{i_{1}}) − (i_{m} + 2i_{m} + ... + 2^{m−1}i_{m}) + 2^{m}i_{m} + \frac{1}{2^{n}}
= s (W_{i_{m}}...W_{i_{1}}) − i_{m} (2^{m} − 1) + i_{m}2^{m} + \frac{1}{2^{n}}
= s (W_{i_{k}}...W_{i_{m+1}}W_{i_{m}}...W_{i_{1}}) − (i_{k} + ... + i_{m−1}) + i_{m} + \frac{1}{2^{n}}
< s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n−1}} .

Also s (W_{1/2^{n}}W_{2i_{1}}) = \frac{1}{2^{n}} + 2i_{1} > 2i_{1} > s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) and

j_{1} = 2i_{1} < s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n−1}} .

Case 5: There is m ∈ \{2, 3, ..., k\} such that i_{m} < \frac{1}{2^{n}} < \frac{i_{m−1}}{2} . Then

W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}} ⊂ W_{1/2^{n}}W_{2i_{m}}W_{i_{m−1}}...W_{i_{1}}
⊂ W_{1/2^{n}}W_{1/2^{n}}W_{i_{m−1}}...W_{i_{1}}
⊂ W_{1/2^{n−1}}W_{i_{m−1}}...W_{i_{1}}.

and

s (W_{1/2^{n−1}}W_{i_{m−1}}...W_{i_{1}}) = s (W_{i_{k}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}}) − (i_{k} + ... + i_{m}) + \frac{1}{2^{n−1}}
< s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n−1}} .

Moreover i_{k} + ... + i_{m} ≤ (\frac{1}{2^{k−m}} + \frac{1}{2^{k−m}} + ...\frac{1}{2} + 1) i_{m} < 2i_{m} ≤ \frac{1}{2^{n}} < \frac{1}{2^{n−1}} . Consequently, s (W_{1/2^{n−1}}W_{i_{m−1}}...W_{i_{1}}) > s (W_{i_{k}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}}).

Case 6: There is m ∈ \{2, 3, ..., k\} such that i_{m} < \frac{1}{2^{n}} = \frac{i_{m−1}}{2} and there is q ∈ {2, 3, ..., m − 1} such that 4i_{q} ≤ i_{q−1}. If p is the greatest element of the set

{q : 2 ≤ q ≤ m − 1, 4i_{q} ≤ i_{q−1}},

then

W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}} ⊂ W_{1/2^{n}}W_{2i_{m}}W_{i_{m−1}}...W_{i_{1}}
⊂ W_{1/2^{n}}W_{1/2^{n}}W_{i_{m−1}}...W_{i_{1}}
⊂ W_{1/2^{n−1}}W_{i_{m−1}}...W_{i_{1}}
⊂ W_{2i_{p}}W_{i_{p−1}}...W_{i_{1}}.

Moreover

s (W_{2i_{p}}W_{i_{p−1}}...W_{i_{1}}) = s (W_{i_{m−1}}...W_{i_{p}}W_{i_{p−1}}...W_{i_{1}}) − (i_{m−1} + .. + i_{p}) + 2i_{p}

= s (W_{i_{m−1}}...W_{i_{p}}W_{i_{p−1}}...W_{i_{1}}) − (i_{m−1} + 2i_{m−1} + ... + 2^{m−p−1}i_{m−1}) + 2^{m−p}i_{m−1}
= s (W_{i_{m−1}}...W_{i_{p}}W_{i_{p−1}}...W_{i_{1}}) − i_{m−1} (2^{m−p} − 1) + i_{m−1}2^{m−p}
= s (W_{i_{k}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}}) − (i_{k} + ... + i_{m}) + i_{m−1}
= s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) − (i_{k} + ... + i_{m}) + \frac{1}{2^{n−1}} .

Hence

s (W_{2i_{p}}W_{i_{p−1}}...W_{i_{1}}) < s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n−1}} .

Since we have i_{k} + ... + i_{m} ≤ (\frac{1}{2^{k−m+1}} + \frac{1}{2^{k−m}} + ...\frac{1}{2}) i_{m−1} < i_{m−1}, it follows that i_{m−1} − (i_{k} + ... + i_{m}) > 0. Thus s (W_{2i_{p}}W_{i_{p−1}}...W_{i_{1}}) > s (W_{i_{k}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}}). We also have y_{r} = 2i_{p} ≥ \frac{1}{2^{n−1}} .

Case 7: There is m ∈ \{2, 3, ..., k\} such that i_{m} < \frac{1}{2^{n}} = \frac{i_{m−1}}{2} = \frac{i_{m−2}}{2^{2}} = ... = \frac{i_{1}}{2^{m−1}} . Then W_{1/2^{n}}W_{i_{k}}...W_{i_{1}} ⊂ W_{1/2^{n}}W_{2i_{m}}W_{i_{m−1}}...W_{i_{1}} ⊂ W_{1/2^{n−1}}W_{i_{m−1}}...W_{i_{1}}W_{1/2^{n}}W_{2i_{1}} ⊂ ...W_{2i_{1}} and

s (W_{2i_{1}}) = s (W_{i_{m−1}}...W_{i_{1}}) − (i_{m−1} + .. + i_{1}) + 2i_{1}

= s (W_{i_{m−1}}...W_{i_{1}}) − (i_{m−1} + 2i_{m−1} + ... + 2^{m−2}i_{m−1}) + 2^{m−1}i_{m−1}
= s (W_{i_{m−1}}...W_{i_{1}}) − i_{m−1} (2^{m−1} − 1) + i_{m−1}2^{m−1} + \frac{1}{2^{n}}
= s (W_{i_{k}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}}) − (i_{k} + ... + i_{m}) + i_{m−1}
< s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n−1}} .

Also s (W_{2i_{1}}) = 2i_{1} > (\frac{1}{2^{k−1}} + ... + \frac{1}{2} + 1) i_{1} ≥ i_{k} + i_{k−1} + ...i_{1} ≥ s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}). Moreover j_{1} = 2i_{1} = s (W_{2i_{1}}) < s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n−1}} .

Case 8: \frac{1}{2^{n}} = i_{1}. We have

W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{1}} ⊂ W_{1/2^{n}}W_{2i_{1}},

s (W_{1/2^{n}}W_{2i_{1}}) = \frac{1}{2^{n}} + 2i_{1} ≤ s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n}} + i_{1}
= s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n−1}} .

and \frac{1}{2^{n}} + 2i_{1} > 2i_{1} > (\frac{1}{2^{k−1}} + ... + \frac{1}{2} + 1) i_{1} ≥ i_{k} + i_{k−1} + ...i_{1} = s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}). We also have j_{1} = 2i_{1} < s (W_{1/2^{n}}W_{2i_{1}}) ≤ s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n−1}} .

Case 9: \frac{1}{2^{n}} > i_{1}. We have

W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{1}} ⊂ W_{1/2^{n}}W_{2i_{1}} ⊂ W_{1/2^{n}}W_{1/2^{n}} ⊂ W_{1/2^{n−1}},

s (W_{1/2^{n−1}}) = \frac{1}{2^{n−1}} < s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n−1}}

and \frac{1}{2^{n−1}} > 2i_{1} > (\frac{1}{2^{k−1}} + ... + \frac{1}{2} + 1) i_{1} ≥ i_{k} + i_{k−1} + ...i_{1} = s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}). Moreover j_{1} = \frac{1}{2^{n−1}} < s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n−1}} .

Let us also remark that if \frac{1}{2^{n}} = i_{k} , then W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{1}} ⊂ W_{2i_{m}}W_{i_{m−1}}...W_{i_{1}}, where m is the greatest element of the set {q : 2 ≤ q ≤ k, 4i_{q} ≤ i_{q−1}} if the set is not empty or m = 1, otherwise. We have

s (W_{2i_{m}}W_{i_{m−1}}...W_{i_{1}}) = s (W_{i_{k}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}}) − (i_{k} + ... + i_{m}) + 2i_{m}
= s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) − (1 + 2 + ... + 2^{k−m}) \frac{1}{2^{n}} + \frac{2^{k−m+1}}{2^{n}}
= s (W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}) + \frac{1}{2^{n}}.

Moreover i_{k} + ... + i_{m} ≤ (\frac{1}{2^{k−m}} + \frac{1}{2^{k−m}} + ...\frac{1}{2} + 1) i_{m} < 2i_{m}. Consequently,

s (W_{2i_{m}}W_{i_{m−1}}...W_{i_{1}}) > s (W_{i_{k}}...W_{i_{m}}W_{i_{m−1}}...W_{i_{1}}).

2.4. Remark. In the preceding lemma since W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{1}} ⊂ W_{j_{r}}W_{j_{r−1}}...W_{j_{1}}, it follows that (W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{1}})^{−1} ⊂ (W_{j_{r}}W_{j_{r−1}}...W_{j_{1}})^{−1} and consequently,

W_{i_{1}}W_{i_{2}}...W_{i_{k}}W_{1/2^{n}} ⊂ W_{j_{1}}W_{j_{2}}...W_{j_{r}}.

A URYSOHN TYPE LEMMA FOR GROUPOIDS  979

2.5. Theorem. Let G be a groupoid, W be a G-uniformity (in the sense of Definition 2.1) and let W ∈ W. Let us consider an I = \{\frac{1}{2^{n}} , n ∈ ℕ\}-indexed subfamily W_{I} = \{W_{i}\}_{i∈I} of W as in Lemma 2.3 such that W_{1} ⊂ W. Then for every subset A of G there is a function f = f_{A,W_{I}} : G → [0, 1] satisfying the following conditions:

1. If n ∈ ℕ, n ≥ 2, x ∈ G and y ∈ W_{1/2^{n}}xW_{1/2^{n}}, then |f (x) − f (y)| < \frac{1}{2^{n−2}} .

2. f (x) = 0 for all x ∈ A.

3. f (x) = 1 for all x ∉ W AW.

4. If A = A^{−1}, then f (x) = f (x^{−1}) for all x ∈ G.

5. If G is endowed with a topology such that W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}} is open for all i_{1}, i_{2}, ..., i_{k} ∈ I, i_{k} < i_{k−1} < ... < i_{1} < 1, then f is upper semi-continuous.

6. For all n ∈ ℕ, n ≥ 2, we have

W_{1/2^{n+1}}AW_{1/2^{n+1}} ⊂ \{x : f (x) < \frac{1}{2^{n}} \} ⊂ W_{1/2^{n−1}}AW_{1/2^{n−1}}.

In particular, if A = G^{(0)}, then

W_{1/2^{n+1}}W_{1/2^{n+1}} ⊂ \{x : f (x) < \frac{1}{2^{n}} \} ⊂ W_{1/2^{n−1}}W_{1/2^{n−1}} ⊂ W_{1/2^{n−2}}

for all n ∈ ℕ, n ≥ 2.

7. If A = G^{(0)}, then f (xy) ≤ 3f (x) + f (y) for all (x, y) ∈ G^{(2)}.

8. If A = G^{(0)}, then f (xy) ≤ 2 (f (x) + f (y)) for all (x, y) ∈ G^{(2)}.

9. If A = G^{(0)}, then f (x_{1}x_{2}...x_{n}) ≤ 3 (f (x_{1}) + f (x_{2}) + ... + f (x_{n})) for all n ∈ ℕ and x_{1}, x_{2}, ..., x_{n} ∈ G such that d (x_{i}) = r (x_{i+1}) for all i ∈ {1, 2, ..., n − 1}.

10. If A = G^{(0)} and for every x ∈ G \ G^{(0)} there is i_{x} ∈ I such that x ∉ W_{i_{x}} (or equivalently, ⋂_{n} W_{1/2^{n}} = G^{(0)}), then f^{−1} ({0}) = G^{(0)}.

Proof. For each x ∈ G, let us define

i (x) = inf \{ \begin{matrix}i_{k} + i_{k−1} + ... + i_{1} : i_{1}, i_{2}, ..., i_{k} ∈ I, i_{k} < i_{k−1} < ... < i_{1}, \\ x ∈ W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}} \end{matrix} \}

(with convention inf ∅ = ∞) and

f (x) = min {i (x) , 1}.

1. Let x ∈ G and y ∈ W_{1/2^{n}}xW_{1/2^{n}}. If i (x) ≥ 1 and i (y) ≥ 1, then f (x) = f (y) = 1. Let us suppose that i (x) < 1 or i (y) < 1.

Case 1: i (x) < 1. Then there are i_{1}, i_{2}, ..., i_{k} ∈ I, i_{k} < i_{k−1} < ... < i_{1} < 1 such that x ∈ W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}} and i_{k} + i_{k−1} + ... + i_{1} < i (x) + \frac{1}{2^{n}}. By Lemma 2.3, there are j_{1}, j_{2}, ..., j_{r} ∈ I, j_{r} < j_{r−1} < i_{r−2} < ... < j_{1} ≤ 1 such that W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{1}} ⊂ W_{j_{r}}W_{j_{r−1}}...W_{j_{1}} and

0 < (j_{r} + ... + j_{1}) − (i_{k} + i_{k−1} + ... + i_{1}) < \frac{3}{2^{n}}

Hence

i_{k} + i_{k−1} + ... + i_{1} ≤ j_{r} + ... + j_{1} < i (x) + \frac{1}{2^{n−2}}

and since

y ∈ W_{1/2^{n}}xW_{1/2^{n}} ⊂
⊂ W_{1/2^{n}}W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}AW_{i_{1}}...W_{i_{k−1}}W_{i_{k}}W_{1/2^{n}}
⊂ W_{j_{r}}W_{j_{r−1}}...W_{j_{1}}AW_{j_{1}}W_{j_{2}}...W_{j_{r}}

it follows that i (y) < i (x) + \frac{1}{2^{n−2}} . If i (y) < 1, then since y ∈ W_{1/2^{n}}xW_{1/2^{n}} is equivalently to x ∈ W_{1/2^{n}}yW_{1/2^{n}} it follows that i (x) < i (y) + \frac{1}{2^{n−2}} . Therefore |f (x) − f (y)| = |i (x) − i (y)| < \frac{1}{2^{n−2}} . If i (y) ≥ 1, then |f (x) − f (y)| = |i (x) − 1| = 1 − i (x) ≤ i (y) − i (x) < \frac{1}{2^{n−2}} .

Case 2: i (y) < 1. Since y ∈ W_{1/2^{n}}xW_{1/2^{n}} is equivalently to x ∈ W_{1/2^{n}}yW_{1/2^{n}}, the case i (y) < 1 can be treated similarly as the case i (x) < 1.

2. Let us prove that f (x) = 0 for all x ∈ A. Since A ⊂ W_{1/2^{n}}AW_{1/2^{n}} for all n, it follows that i (x) = 0, and consequently, f (x) = 0 for all x ∈ A.

3. Let us prove that f (x) = 1 for all x ∉ W AW. Let x ∉ W AW. By contradiction, let us suppose f (x) < 1. We necessarily have i (x) < 1, and hence there are i_{1}, i_{2}, ..., i_{k} ∈ I, i_{k} < i_{k−1} < ... < i_{1} < 1 such that

x ∈ W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}} ⊂ W_{2i_{1}}AW_{2i_{1}}
⊂ W_{1}AW_{1} ⊂ W AW

This is in contradiction to the hypothesis x ∉ W AW.

4. Since A = A^{−1}, it follows that

(W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}})^{−1} = W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}}.

Thus x ∈ W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}} if and only if

x^{−1} ∈ W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}}.

Therefore f (x) = f (x^{−1}) for all x ∈ G.

5. Let α ∈ ℝ and let us consider the set

U_{α} = {x ∈ G : f (x) < α}.

If α > 1, then U_{α} = G, hence U_{α} is an open set. Let us consider α ≤ 1 and let x ∈ U_{α}. Then f (x) < 1. Thus i (x) < 1, and hence there are i_{1}, i_{2}, ..., i_{k} ∈ I, i_{k} < i_{k−1} < ... < i_{1} < 1 such that

x ∈ W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}}
i_{k} + i_{k−1} + ... + i_{1} < α.

For all y ∈ W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}} we have i (y) < α. Consequently,

x ∈ W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}} ⊂ U_{α}.

Therefore U_{α} is open.

6. If x ∈ W_{1/2^{n+1}}AW_{1/2^{n+1}}, then i (x) ≤ \frac{1}{2^{n+1}} . Thus f (x) ≤ \frac{1}{2^{n+1}} < \frac{1}{2^{n}}. If f (x) < \frac{1}{2^{n}} < 1, then i (x) < \frac{1}{2^{n}} and there are i_{1}, i_{2}, ..., i_{k} ∈ I, i_{k} < i_{k−1} < ... < i_{1} < 1 such that x ∈ W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}A W_{i_{1}}...W_{i_{k−1}}W_{i_{k}} and i_{k} + i_{k−1} + ... + i_{1} < i (x) + \frac{1}{2^{n}} < \frac{1}{2^{n−1}} . Hence i_{1} < \frac{1}{2^{n−1}} and therefore x ∈ W_{2i_{1}}AW_{2i_{1}} ⊂ W_{1/2^{n−1}}AW_{1/2^{n−1}}.

7. Let (x, y) ∈ G^{(2)}. If 3f (x) + f (y) ≥ 1, then obviously, f (xy) ≤ 3f (x) + f (y). Let us suppose that 3f (x) + f (y) < 1 or equivalently, 3i (x) + i (y) < 1 (consequently, i (x) < \frac{1}{3} and i (y) < 1). Let ε > 0 such that ε < 1 − 3i (x) − i (y). Then there are i_{1}, i_{2}, ..., i_{k} ∈ I, i_{k} < i_{k−1} < ... < i_{1} ≤ \frac{1}{4} such that x ∈ W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}W_{i_{1}}...W_{i_{k−1}}W_{i_{k}}, i_{k} + i_{k−1} + ... + i_{1} < i (x) + \frac{ε}{3} and there are j_{1}, j_{2}, ..., j_{m} ∈ I, j_{m} < j_{m−1} < ... < j_{1} ≤ \frac{1}{2} such that y ∈ W_{j_{m}}W_{j_{m−1}}...W_{j_{1}}W_{j_{1}}...W_{j_{m−1}}W_{j_{m}}, j_{m} + j_{m−1} + ... + j_{1} < i (y) + \frac{ε}{3}. By Lemma 2.3, there are q_{1}^{1}, q_{2}^{1}, ..., q_{r_{1}}^{1} ∈ I, q_{r_{1}}^{1} < q_{r_{1}−1}^{1} < q_{r_{1}−2}^{1} < ... < q_{1}^{1} ≤ 1 such that W_{i_{k}}W_{j_{m}}W_{i_{m−1}}...W_{j_{1}} ⊂ W_{q_{r_{1}}^{1}}W_{q_{r_{1}−1}^{1}}...W_{q_{1}^{1}},

0 < (q_{r_{1}}^{1} + ... + q_{1}^{1}) − (j_{m} + j_{m−1} + ... + j_{1}) ≤ 2i_{k}.

and q_{1}^{1} ≤ j_{m}+j_{m−1}+...+j_{1}+2i_{k} < i (y)+ \frac{ε}{3} +2i (x)+ \frac{2ε}{3} < 1. Repeatedly applying Lemma 2.3, for p = 2, 3, ..., k there are q_{1}^{p}, q_{2}^{p}, ..., q_{rp}^{p} ∈ I, q_{r_{p}}^{p} < q_{r_{p}−1}^{p} < q_{r_{p}−2}^{p} < ... < q_{1}^{p} ≤ 1 such that W_{i_{k−p+1}}W_{q_{r_{p=1}}^{p=1}}W_{q_{r_{p−1}−1}^{p−1}}...W_{q_{1}^{p−1}} ⊂ W_{q_{rp}^{p}}W_{q_{rp−1}^{p}}...W_{q_{1}^{p}} ,

0 ≤ (q_{r_{p}}^{p} + ... + q_{1}^{p}) − (q_{r_{p−1}}^{p−1} + ... + q_{1}^{p−1}) ≤ 2i_{k−p+1}.

and

q_{1}^{p−1} < q_{r_{p−1}}^{p−1} + ... + q_{1}^{p−1} + 2i_{k−p+1}
< q_{r_{p−2}}^{p−2} + ... + q_{1}^{p−2} + 2i_{k−p} + 2i_{k−p+1}
........
< j_{m} + j_{m−1} + ... + j_{1} + 2i_{k} + ... + 2i_{k−p} + 2i_{k−p+1}
< i (y) + \frac{ε}{3} + 2i (x) + \frac{2ε}{3} < 1.

Applying again Lemma 2.3, there are q_{1}^{k+1}, q_{2}^{k+1}, ..., q_{r_{k+1}}^{k+1} ∈ I, q_{r_{k+1}}^{k+1} < q_{r_{k+1}−1}^{k+1} < q_{r_{k+1}−2}^{k+1} < ... < q_{1}^{k+1} ≤ 1 such that W_{i_{1}}W_{q_{r_{k}}^{k}} W_{q_{r_{k}−1}^{k}}...W_{q_{1}^{k}} ⊂ W_{q_{rk+1}^{k+1}}W_{q_{r_{k+1}−1}^{k+1}}...W_{q_{1}^{k+1}} and

0 < (q_{r_{k+1}}^{k+1} + ... + q_{1}^{k+1}) − (q_{r_{k}}^{k} + ... + q_{1}^{k}) < i_{1}

Moreover q_{r_{k+1}}^{k+1} ≥ i_{1}. Hence W_{i_{1}}W_{i_{1}}W_{i_{2}}...W_{i_{k}}W_{j_{m}}W_{i_{m−1}}...W_{j_{1}} ⊂ W_{q_{rk+1}^{k+1}}W_{q_{r_{k+1}−1}^{k+1}}...W_{q_{1}^{k+1}} and

0 < (q_{r_{k+1}}^{k+1} + ... + q_{1}^{k+1}) − (j_{m} + j_{m−1} + ... + j_{1})
< 2 (i_{k} + i_{k−1} + ... + i_{1}) + i_{1}.

Thus W_{i_{k}}...W_{i_{2}}W_{i_{1}}W_{i_{1}}W_{i_{2}}...W_{i_{k}}W_{j_{m}}W_{i_{m−1}}...W_{j_{1}} ⊂ W_{i_{k}}...W_{i_{2}}W_{q_{rk+1}^{k+1}}W_{q_{r_{k+1}−1}^{k+1}}...W_{q_{1}^{k+1}}. Consequently,

xy ∈ W_{i_{k}}...W_{i_{2}}W_{i_{1}}W_{i_{1}}W_{i_{2}}...W_{i_{k}}W_{j_{m}}W_{j_{m−1}}...W_{j_{1}}W_{j_{1}}...W_{j_{m}}
⊂ W_{i_{k}}...W_{i_{2}}W_{q_{rk+1}^{k+1}}W_{q_{r_{k+1}−1}^{k+1}}...W_{q_{1}^{k+1}}W_{q_{1}^{k+1}}...W_{q_{rk+1}^{k+1}}W_{i_{2}}...W_{i_{k}}

and i_{k} < i_{k−1} < ... < i_{2} < q_{r_{k+1}}^{k+1} < q_{r_{k+1}−1}^{k+1} < q_{r_{k+1}−2}^{k+1} < ... < q_{1}^{k+1} ≤ 1. Hence

i (xy) ≤ i_{k} + i_{k−1} + ... + i_{2} + (q_{r_{k+1}}^{k+1} + ... + q_{1}^{k+1})
< i_{k} + i_{k−1} + ... + i_{2} + 2 (i_{k} + i_{k−1} + ... + i_{1}) + i_{1} + (j_{m} + j_{m−1} + ... + j_{1})
≤ 3 (i_{k} + i_{k−1} + ... + i_{1}) + (j_{m} + j_{m−1} + ... + j_{1})
< 3i (x) + i (y) + \frac{4}{3}ε

for all ε > 0. Therefore i (xy) ≤ 3i (x) + i (y) for all (x, y) ∈ G^{(2)}. Thus f (xy) ≤ 3f (x) + f (y) for all (x, y) ∈ G^{(2)}.

8. Let (x, y) ∈ G^{(2)}. We proved in 7 that

f (xy) ≤ 3f (x) + f (y) .

On the other hand we have f (x^{−1}) = f (x), f (y^{−1}) = f (y) and

f (xy) = f (y^{−1}x^{−1}) ≤_{7} 3f (y^{−1}) + f (x^{−1}) = 3f (y) + f (x) .

Adding the last inequalities we obtain

2f (xy) ≤ 3 (f (x) + f (y)) + f (x) + f (y) = 4 (f (x) + f (y)) .

9. We prove the inequality by mathematical induction. For n = 2 is true, since by 7 we have f (x_{1}x_{2}) ≤ 3f (x_{1}) + f (x_{2}) ≤ 3f (x_{1}) + 3f (x_{2}). Let us suppose that the inequality is true for some n and let us prove that it is true for n + 1. Using 7 we obtain

f (x_{1}x_{2}...x_{n}x_{n+1}) ≤_{7} 3f (x_{1}) + f (x_{2}...x_{n}x_{n+1})

≤ 3f (x_{1}) + 3 (f (x_{2}) + f (x_{3}) + ... + f (x_{n})) .

10. If x ∈ G^{(0)} then by 2, f (x) = 0. Conversely, if f (x) = 0, then for all n, we have i (x) < \frac{1}{2^{n}}. Thus there are i_{1}, i_{2}, ..., i_{k} ∈ I, i_{k} < i_{k−1} < ... < i_{1} ≤ \frac{1}{2^{n}} such that x ∈ W_{i_{k}}W_{i_{k−1}}...W_{i_{1}}W_{i_{1}}...W_{i_{k−1}}W_{i_{k}} and i_{k} + i_{k−1} + ... + i_{1} < \frac{1}{2^{n}}. Since W_{i_{k}}W_{i_{k−1}}...W_{i_{1}} ⊂ W_{2i_{1}},

it follows that x ∈ W_{4i_{1}} ⊂ W_{1/2^{n−2}} for all n ≥ 2. Thus x ∈ ⋂_{n} W_{1/2^{n}} = G^{(0)}.

2.6. Proposition. Let G be a groupoid, W be a G-uniformity and f : G → [0, 1] be a function satisfying conditions 2, 4, 9 and 10 in Theorem 2.5 (f associated to A = G^{(0)}). Then there is a function f_{reg} : G → [0, 1] satisfying the following conditions:

1. \frac{1}{3}f ≤ f_{reg} ≤ f.

2. f_{reg}(x) = f_{reg}(x^{−1}) for all x ∈ G.

3. f_{reg} (xy) ≤ f_{reg} (x) + f_{reg} (y) for all (x, y) ∈ G^{(2)}.

4. |f_{reg} (sxt) − f_{reg} (x)| ≤ f_{reg} (s) + f_{reg} (t) for all s, t, x ∈ G with x ∈ G_{r(t)}^{d(s)}.

5. W_{1/2^{n+1}} ⊂ W_{1/2^{n+1}}W_{1/2^{n+1}} ⊂ \{x : f_{reg} (x) < \frac{1}{2^{n}} \} ⊂ W_{1/2^{n=3}}W_{1/2^{n−3}} ⊂ W_{1/2^{n−4}} for all n ∈ ℕ, n ≥ 2.

Proof. In the spirit of [8, Theorem 3.26] let us define f_{reg} : G → [0, 1] by

f_{reg} (x) = inf {∑_{i=1}^{n} f (x_{i}) : x_{1}x_{2}...x_{n} = x} for all x ∈ G.

Then f_{reg} obviously satisfies conditions 1 − 3.

4. Let s, t, x ∈ G such that x ∈ G_{r(t)}^{d(s)}. Then f_{reg} (sxt) ≤ f_{reg} (s) + f_{reg} (x) + f_{reg} (t) and consequently, f_{reg} (sxt) − f_{reg} (x) ≤ f_{reg} (s) + f_{reg} (t). On the other hand f_{reg} (x) = f_{reg} (s^{−1}sxtt^{−1}) ≤ f_{reg} (s^{−1}) + f_{reg} (sxt) + f_{reg} (t^{−1}) = f_{reg} (s) + f_{reg} (sxt) + f_{reg} (t) and therefore f_{reg} (x) − f_{reg} (sxt) ≤ f_{reg} (s) + f_{reg} (t).

5. Let x ∈ W_{1/2^{n+1}}W_{1/2^{n+1}}. Then f_{reg} (x) ≤ f (x) < \frac{1}{2^{n}}. Conversely, let x be such that f_{reg} (x) < \frac{1}{2^{n}}. Then \frac{1}{4}f (x) ≤ \frac{1}{3}f (x) ≤ f_{reg} (x) < \frac{1}{2^{n}}. Hence f (x) < \frac{1}{2^{n−2}} and therefore x ∈ W_{1/2^{n−3}}W_{1/2^{n−3}}.

3. A groupoid generalization of Alexandroff–Urysohn Theorem

As we remark in [5, p. 57], if G is a topological groupoid whose unit space is a T_{1}-space (the points are closed in G^{(0)}), then the topologies of the r-fibres, as well as the topologies of the d-fibres, are determined by a fundamental system of neighborhoods {W }_{W ∈W} of G^{(0)}. More precisely, for each u ∈ G^{0} and each x ∈ G^{u} (respectively, x ∈ G_{u}), {xW }_{W ∈W} (respectively, {W x}_{W ∈W} ) is a local basis for x with respect to the topology induced by G on G^{u} (respectively, G_{u}). We also prove in [5, p. 59] that if W satisfies the conditions imposed to a G-uniformity, then there is a topology denoted τ_{W}^{r} (respectively, τ_{W}^{d}) on G such that for all x ∈ G, V^{r} (x) (respectively, V^{d} (x)) is a neighborhood basis for x, where

V^{r} (x) = {V ⊂ G : there is W ∈ W such that xW ⊂ V }.

respectively,

V^{d} (x) = {V ⊂ G : there is W ∈ W such that W x ⊂ V }.

Unlike the case of a group, a groupoid G (that isn’t a group) is generally not a topological groupoid with respect to τ_{W}^{r} or τ_{W}^{d}. That is why we define a new topology associated to a G-uniformity.

3.1. Definition. Let G be a groupoid endowed with a G-uniformity W. The topology τ_{W} induced by the G-uniformity W is the topology on G defined in the following way: A ∈ τ_{W} if and only if for every x ∈ A there is W_{x} ∈ W such that W_{x}xW_{x} ⊂ A.

For each x ∈ G let us write

V (x) = {V ⊂ G : there is W ∈ W such that W xW ⊂ V }.

In order to see that τ_{W} is indeed a topology it is enough to prove that for all V ∈ V (x), there is U ∈ V (x) such that V ∈ V (y) for all y ∈ U. Since V ∈ V (x), it follows that there is W_{x} ∈ W such that W_{x}xW_{x} ⊂ V . Let W′_{x} ∈ W such that W′_{x}W′_{x} ⊂ W_{x}. If we take U = W′_{x}xW′_{x}, then for all y ∈ U there is s ∈ W′_{x} ∩ G^{d(x)} and t ∈ W′_{x} ∩ G_{r(x)}) such that y = txs and

W′_{x}yW′_{x} = W′_{x}txsW′_{x} ⊂ W′_{x}W′_{x}xW′_{x}W′_{x} ⊂ W_{x}xW_{x}.

Alternatively, we can note that τ_{W} is the topology on G induced by the following uniform structure U_{W} associated with the G-uniformity W: U ∈ U_{W} if and only if there is W ∈ W such that {x} × W xW ⊂ U for all x ∈ G.

Let us remark that for two equivalent G-uniformities W and W′ in the sense of Definition 2.2 we have τ_{W} = τ_{W′}.

In [5] we introduced the notions of left uniform continuity on fibres and right uniform continuity on fibres reformulating the definition of left and right uniform continuity [3, Definition 3.1/p. 39] in the setting of a groupoid endowed with a family of subsets satisfying the conditions imposed to a G-uniformity. Let us define a new notion of uniform continuity with respect to a G-uniformity.

3.2. Definition. Let G be a groupoid endowed with a G-uniformity W, A ⊂ G and E be a Banach space. The function h : A → E is said to be uniformly continuous on fibres (with respect to W) if and only if for each ε > 0 there is W_{ϵ} ∈ W such that:

∥h (x) − h (sxt)∥ < ε for all s, t ∈ W_{ε} and x ∈ A ∩ G_{r(t)}^{d(s)} such that sxt ∈ A.

3.3. Remark. The function f defined in Theorem 2.5 as well as the function f_{reg} in Proposition 2.6 are uniformly continuous on fibres with respect to the corresponding Guniformity.

We will prove (Proposition 3.8) that if there is an appropriate connection between the G-uniformity and the topology of G, then the restrictions of a uniformly continuous on fibres function to r-fibres as well as to d-fibres are continuous functions.

3.4. Definition. Let G be a groupoid endowed with a topology τ. Let {W_{j}}_{j∈J} be a collection of subsets of G such that for all j ∈ J, G^{(0)} ⊂ W_{j} and W_{j} = W_{j}^{−1}. The collection {W_{j}}_{j∈J} is said to be compatible with the topology of the r-fibres (respectively, d-fibres) if for every u ∈ G^{(0)} and every open neighborhood U of u, there is j ∈ J such that W_{j} ∩ G^{u} ⊂ U ∩ G^{u} and u is in the interior of W_{j} ∩ G^{u} with respect to the topology on G^{u} coming from (G, τ) (respectively, W_{j} ∩ G_{u} ⊂ U ∩ G_{u} and u is in the interior of W_{j} ∩ G_{u} with respect to the topology on G_{u} coming from (G, τ)).

The collection {W_{j}}_{j∈J} is said to be compatible with the topology of the fibres if it is compatible with the topology of the r-fibres and d-fibres.

3.5. Remark. If G is groupoid endowed with a topology τ such that the inverse map is continuous, then a collection {W_{j}}_{j∈J} is compatible with the topology of the r-fibres if and only if it is compatible with the topology of the d-fibres.

If G is a topological groupoid and G^{(0)} is a T_{1}-space (the points are closed in G^{(0)}), then any fundamental system of symmetric neighborhoods of G^{(0)} is compatible with the topology of the fibres. Indeed, let u ∈ G^{(0)}. Since G^{(0)} is a T_{1}-space, G \ G^{u} is open for all u. If U is an open subset of G containing u, then U ∪ (G \ G^{u}) is an open neighborhood of G^{(0)}. Thus there is W ∈ W such that W ⊂ U ∪ (G \ G^{u}), and W ∩ G^{u} ⊂ U ∩ G^{u}.

If G is a topological groupoid and {W_{j}}_{j∈J} is compatible with the topology of the r-fibres (and hence to d-fibres), then the topologies of the r-fibres and d-fibres are determined by {W_{j}}_{j∈J}: for each u ∈ G^{0} and each x ∈ G^{u} (respectively, x ∈ G_{u}), {xW_{j}}_{j∈J} (respectively, {W_{j}x}_{j∈J} ) is a local basis for x with respect to the topology induced by G on G^{u} (respectively, G_{u}).

3.6. Proposition. If G is a groupoid endowed with a topology such that for all x ∈ G the map y ↦ xyx^{−1} [: G_{d(x)}^{d(x)} → G_{r(x)}^{r(x)}] is continuous at d (x) and if W is compatible with the topology of the r- fibres or d-fibres, then for every W_{1} ∈ W and x ∈ G there is W_{2} ∈ W such that W_{2} ∩ G_{d(x)}^{d(x)} ⊂ x^{−1}W_{1}x (or equivalently, xW_{2}x^{−1} ⊂ W_{1}).

Proof. Let W_{1} ∈ W and x ∈ G. Since xr (x) x^{−1} ∈ W_{1} ∩ G_{r(x)}^{r(x)}, it follows that there is

an open neighborhood V of d (x) such that xV x^{−1} ⊂ W_{1} ∩ G_{r(x)}^{r(x)}. Let W_{2} ∈ W such that W_{2} ∩ G^{d(x)} ⊂ V ∩ G^{d(x)} or W_{2} ∩ G_{d(x)} ⊂ V ∩ G_{d(x)}. Then xW_{2}x^{−1} ⊂ xV x^{−1} ⊂ W_{1}.

A topological groupoid is said to be locally transitive (see [12]) if for all u ∈ G^{(0)} the maps r_{u} are open, where r_{u} : G_{u} → G^{(0)} is defined by r_{u} (x) = r (x) for all x ∈ G_{u} and G_{u} is endowed with the topology coming from G. Hence the maps d_{u} are open, where d_{u} : G^{u} → G^{(0)}, d_{u} (x) = d (x) for all x ∈ G^{u} and G^{u} is endowed with the topology coming from G. Topological groups and pair groupoids X × X (X topological space) are topological locally transitive groupoids. More general any trivial groupoid X × G × X (X topological space and G topological group) is locally transitive. Any transitive Polish groupoid with open range map is locally transitive [10] (see [9, p. 8] for transitive locally compact second countable groupoids with open range maps).

3.7. Proposition. Let G be a groupoid and W be a G-uniformity such that for every W_{1} ∈ W and x ∈ G there is W_{2} ∈ W such that W_{2} ∩ G_{d(x)}^{d(x)} ⊂ x^{−1}W_{1}x (or equivalently, xW_{2}x^{−1} ⊂ W_{1}). Then G is a topological locally transitive groupoid with respect to the topology τ_{W} induced by the G-uniformity W (in the sense of Definition 3.1). The topologies τ_{W}^{r} and τ_{W}^{d} are finer than τ_{W}. However the topologies induced by τ_{W}^{r} and τ_{W} on r-fibres (respectively, by τ_{W}^{d} and τ_{W} on d-fibres) coincide.

Proof. Let us show that the inversion map and the product map are continuous with respect to τ_{W}. The fact that (W xW)^{−1} = W x^{−1}W (x ∈ G and W ∈ W) implies that the inversion is a homeomorphism. For all W ∈ W, there is W_{1} ∈ W such that W_{1}W_{1} ⊂ W and for all y ∈ G there is W_{y} ∈ W such that W_{y} ⊂ W and W_{y} ∩ G_{r(y)}^{r(y)} ⊂ yW_{1}y^{−1} or equivalently, y^{−1}W_{y}y ⊂ W_{1}. If Wy^{′} ∈ W is such that Wy^{′}Wy^{′} ⊂ W_{y} and x ∈ G_{r(y)}, then

Wy^{′}xWy^{′}Wy^{′}yWy^{′} ⊂ Wy^{′}xyy^{−1}W_{y}yWy^{′} ⊂ Wy^{′}xyW_{1}Wy^{′} ⊂ W xyW,

Therefore the product map is continuous.

Obviously, the topologies τ_{W}^{r} and τ_{W}^{d} are finer than τ_{W} (xW ⊂ W xW and W x ⊂ W xW). For every u ∈ G^{(0)}, x ∈ G^{u} and W ∈ W there is W_{1} ∈ W such that W_{1}W_{1} ⊂ W and there is W_{x} ∈ W such that W_{x} ⊂ W_{1} and W_{x} ∩ G_{r(x)}^{r(x)} ⊂ xW_{1}x^{−1} or equivalently, x^{−1}W_{x}x ⊂ W_{1}. Thus

W_{x}xW_{x} ∩ G^{u} = xx^{−1}W_{x}xW_{x} ⊂ xW_{1}W_{x} ⊂ xW.

Hence the topologies induced by τ_{W}^{r} and τ_{W} on r-fibres coincide. Similarly, the topologies induced by τ_{W}^{d} and τ_{W} on d-fibres coincide.

Let u ∈ G^{(0)}. In order to prove that d_{u} : G^{u} → G^{(0)} is open it suffices to note that if x ∈ G^{u} and W ∈ W, then

d_{u} (G^{u} ∩ W xW) = d_{u} (G^{d(x)} ∩ W ) = (W d (x) W) ∩ G^{(0)}.

3.8. Proposition. Let (G, τ) be a topological groupoid and W be a G-uniformity compatible with the topology of the fibres. Then:

1. The topology τ_{W} (induced by the G-uniformity W) is finer than τ (the original topology of G).

2. The topologies induced by τ and τ_{W} on r-fibres (respectively, on d-fibres) coincide.

3. If (G, τ) is locally transitive, then the topology τ_{W} coincides with τ on G.

Proof. 1. Let U be an open subset of G with respect to τ and let x ∈ U. Since xd (x) ∈ U, it follows that there is an open neighborhood U_{1} ∈ τ of x and an open neighborhood V_{1} ∈ τ of d (x) such that U_{1}V_{1} ⊂ U. Moreover since r (x) x ∈ U_{1}, it follows that there is an open neighborhood V_{2} ∈ τ of r (x) such that V_{2}x ⊂ U_{1}. Hence V_{2}xV_{1} ⊂ U. Let W_{1} ∈ W such that W_{1} ∩ G^{d(x)} ⊂ V_{1} ∩ G^{d(x)}, W_{2} ∈ W such that W_{2} ∩ G_{r(x)} ⊂ V_{2} ∩ G_{r(x)} and let W ∈ W such that W ⊂ W_{1} ∩ W_{2}. Then W xW ⊂ V_{2}xV_{1} ⊂ U. Thus U is open with respect to τ_{W}.

2. Since τ_{W} is finer than τ, it suffices to prove that for all u ∈ G^{(0)}, x ∈ G^{u} (respectively, x ∈ G_{u}) and all W ∈ W, W xW ∩ G^{u} (respectively, W xW ∩ G_{u}) is a neighborhood of x with respect to the topology on G^{u} (respectively, G_{u}) induced by τ. Since the map y ↦ xy [: G^{d(x)} → G^{r(x)}] (respectively, y ↦ yx [: G_{r(x)} → G_{d(x)}]) is a homeomorphism (with respect to τ), it follows that x (W ∩ G^{d(x)}) = xW (respectively, (W ∩ G_{r(x)}) x = W x) is a neighborhood of x in G^{r(x)} (respectively, G_{d(x)}) with respect to the topology induced by τ. Therefore W xW ∩ G^{u} ⊃ xW (respectively, W xW ∩ G_{u} ⊃ W x) is a neighborhood of x with respect to the topology on G^{u} (respectively, G_{u}) induced by τ.

3. Let us assume that (G, τ) is locally transitive, or equivalently, that for all u ∈ G^{(0)}, d_{u} : G^{u} → G^{(0)} (d_{u} (x) = d (x)) is open. Since τ_{W} is finer than τ, in order to show that τ_{W} = τ it suffices to prove for all x ∈ G and all W ∈ W, x is in the interior of W xW with respect to τ. For each u ∈ G^{(0)} let W^{u} be the interior of G^{u} ∩ W seen as a subset of the topological space G^{u} and let W_{0} = ⋃_{u∈G^{(0)}} W^{u} and W_{1} = ⋃_{u∈G^{(0)}} (W^{u})^{−1}.

Then G^{(0)} ⊂ W_{0} ⊂ W and G^{(0)} ⊂ W_{1} ⊂ W. We prove that W_{1}xW_{0} is open with respect to τ. Let s ∈ W_{1}, t ∈ W_{0} and (y_{i})_{i} be a net in G converging to sxt (with respect to τ). Then d (y_{i}) → d (t) = d_{r(t)} (t). Since d_{r(t)} : G^{r(t)} → G^{(0)} is open, we may pass to a subnet and assume that there are t_{i} ∈ G^{r(t)} such that t_{i} → t and d (t_{i}) = d (y_{i}) for all i. If s_{i} = y_{i}t_{i}^{−1}x^{−1}, then s_{i} → sxtt^{−1}x^{−1} = s. Since t_{i} → t ∈ W_{0}∩G^{r(t)} and s_{i} → s ∈ W_{1}∩G_{r(x)}, it follows that s_{i} are eventually in W_{1} and t_{i} are eventually in W_{0}. Therefore y_{i} = s_{i}xt_{i} is eventually in W_{1}xW_{0}. Thus x is in the interior of W xW.

3.9. Proposition. Let G be a groupoid endowed with a pseudometric ρ satisfying the following conditions:

1. ρ (x, r (x)) = ρ (x^{−1}, d (x)) for all x ∈ G.

2. ρ (xy, r (x)) ≤ ρ (y, r (y)) + ρ (x^{−1}, d (x)) for all (x, y) ∈ G^{(2)}.

For every n ∈ ℕ let

W_{n} := \{x ∈ G : ρ (x, r (x)) < \frac{1}{2^{n}} \}.

Then W = {W_{n}}_{n} is a G-uniformity compatible with the topology of r-fibres (induced by the pseudometric ρ).

Proof. Obviously, satisfies condition 1, 2 and 4 from Definition 2.1. Also let us note that W_{n+1}W_{n+1} ⊂ W_{n} for all n (since ρ (xy, r (x)) ≤ ρ (y, r (y)) + ρ (x^{−1}, d (x)) = ρ (y, r (y)) + ρ (x, r (x)) for all (x, y) ∈ G^{(2)}). Since for all u, W_{n} ∩ G^{u} = B (u, \frac{1}{2^{n}} ) ∩ G^{u}, it follows that W is compatible with the topology of r-fibres.

3.10. Proposition. Let G be a groupoid endowed with a pseudometric ρ satisfying the following conditions:

1. ρ (x, d (x)) = ρ (x^{−1}, r (x)) for all x ∈ G.

2. ρ (xy, d (y)) ≤ ρ (x, d (x)) + ρ (y^{−1}, r (y)) for all (x, y) ∈ G^{(2)}.

For every n ∈ ℕ let

W_{n} := \{x ∈ G : ρ (x, d (x)) < \frac{1}{2^{n}} \}.

Then W = {W_{n}}_{n} is a G-uniformity compatible with the topology of d-fibres (induced by the pseudometric ρ).

Proof. The proof is similar to the proof of Proposition 3.9.

3.11. Definition. Let G be a groupoid endowed with a pseudometric ρ satisfying conditions 1 and 2 in Proposition 3.9 or in Proposition 3.10. Then the G-uniformity constructed in Proposition 3.9 as well as the G-uniformity constructed in Proposition 3.10 will be called the G-uniformity associated to the pseudometric ρ.

3.12. Remark. If ρ is a left invariant pseudometric on a groupoid G (in the sense that ρ (zx, zy) = ρ (x, y) for all x, y, z ∈ G with d (z) = r (x) = r (y)), then ρ (xy, r (x)) ≤ ρ (x^{−1}, d (x))+ρ (y, r (y)) for all (x, y) ∈ G^{(2)} and ρ (x, r (x)) = ρ (x^{−1}, d (x)) for all x ∈ G. Indeed, ρ (xy, r (x)) = ρ (xy, xx^{−1}) = ρ (y, x^{−1}) ≤ ρ (y, r (y))+ρ (x^{−1}, r (y)) = ρ (y, r (y))+ ρ (x^{−1}, d (x)) for all (x, y) ∈ G^{(2)}. Also ρ (x, r (x)) = ρ (xd (x) , xx^{−1}) = ρ (d (x) , x^{−1}) for all x ∈ G.

Also if ρ is a right invariant pseudometric on a groupoid G (in the sense that ρ (xz, yz) =ρ (x, y) for all x, y, z ∈ G with r (z) = d (x) = d (y)), then ρ (xy, d (x)) ≤ ρ (x, d (x)) + ρ (y^{−1}, r (y)) for all (x, y) ∈ G^{(2)}and ρ (x, d (x)) = ρ (x^{−1}, r (x)) for all x ∈ G.

3.13. Remark. Any topological groupoid that is paracompact admits a fundamental system W of neighborhoods that is a G-uniformity compatible with the topology of fibres [10]. The same is true for a topological groupoid with paracompact unit space [5].

3.14. Proposition. Let G be a groupoid and W = {W_{n}}_{n∈ℕ} be a countable G-uniformity. Then G can be endowed with a pseudometric ρ satisfying the following conditions:

1. ρ is left invariant in the sense that ρ (zx, zy) = ρ (x, y) for all x, y, z ∈ G with d (z) = r (x) = r (y).

2. ρ induces a G-uniformity equivalent to W.

3. For every u ∈ G^{(0)} the restriction of ρ to G^{u} is compatible with the topology induced by τ_{W}^{r} on G^{u}.

4. If ⋂_{n∈ℕ} W_{n} = G^{(0)}, then ρ is a metric.

Proof. Let I = \{\frac{1}{2^{n}} , n ∈ ℕ\}. Let W_{0} ∈ \{W_{n}\}_{n∈ℕ} and W′_{1} ∈ W be such that W′_{1}W′_{1} ⊂ W_{0} and W′_{1} ⊂ W_{1}. Inductively we construct an I-indexed family {W′_{i}}_{i∈I}. Suppose that for W′_{1/2^{n}} ∈ W has already been built. Then there is a W′′ ∈ W such that W′′W′′ ⊂ W′_{1/2^{n}} and W′′ ⊂ W_{n+2}. Let W′_{1/2^{n+1}} = W′′. Thus we obtain an I-indexed family W′ = {W′_{i}}_{i∈I} as in Theorem 2.5 and if ⋂_{n∈ℕ} W_{n} = G^{(0)}, then G^{(0)} = ⋂_{i∈I} W′_{i}. Let f = f_{G^{(0)},W′} be the

function defined in Theorem 2.5 and f_{reg} the function associated to f in Proposition 2.6. Thus as in [8] we may define the following distance ρ (x, y) = f_{reg} (x^{−1}y) if r (x) = r (y) and ρ (x, y) = 1 otherwise. Let n ∈ ℕ, n ≥ 4 and u ∈ G^{(0)}. For x ∈ G^{u} we have

B (x, \frac{1}{2^{n}} ) = \{y ∈ G^{u} : f_{reg} (x^{−1}y) < \frac{1}{2^{n}} \}
⊂ xW′_{1/2^{n−4}}

On the other hand according Proposition 2.6 W′_{1/2^{n+1}} ⊂ \{z : f_{reg} (z) < \frac{1}{2^{n}} \}. Hence xW′_{1/2^{n+1}} ⊂ \{y : f_{reg} (x^{−1}y) < \frac{1}{2^{n}} \} = B (x, \frac{1}{2^{n}} ). Thus the topologies induced by τ_{W}^{r} and the metric ρ|_{G^{u}} on G^{u} coincide.

3.15. Proposition. Let G be a groupoid and W = {W_{n}}_{n∈ℕ} be a countable G-uniformity. Then G can be endowed with a pseudometric ρ satisfying the following conditions:

1. ρ is right invariant in the sense that ρ (xz, yz) = ρ (x, y) for all x, y, z ∈ G with r (z) = d (x) = d (y).

2. ρ induces a G-uniformity equivalent to W.

3. For every u ∈ G^{(0)} the restriction of ρ to G_{u} is compatible with the topology induced by τ_{W}^{d} on G_{u}.

4. If ⋂_{n∈ℕ} W_{n} = G^{(0)}, then ρ is a metric.

Proof. Similar as in the proof of Proposition 3.14 we may define the following distance ρ (x, y) = f_{reg} (xy^{−1}) if d (x) = d (y) and ρ (x, y) = 1 otherwise.

3.16. Theorem. A topological locally transitive groupoid. The following statements are equivalent:

(a) G is metrizable

(b) G is paracompact and G^{(0)} has a countable fundamental system {W_{n}}_{n∈ℕ} of neighbor-

hoods such that ⋂_{n∈ℕ} W_{n} = G^{(0)} and ⋂_{n∈ℕ} (r, d) (W_{n}) = diag (G^{(0)}).

(c) For every neighborhood W of G^{(0)} there is a neighborhood W′ of G^{(0)} such that W′W′ ⊂ W and G^{(0)} has a countable fundamental system {W_{n}}_{n∈ℕ} of neighborhoods such that

⋂_{n∈ℕ} W_{n} = G^{(0)} and ⋂_{n∈ℕ} (r, d) (W_{n}) = diag (G^{(0)}).

(d) There is a countable G-uniformity {W_{n}}_{n∈ℕ} compatible with the topology of the fibres such that ⋂_{n∈ℕ} W_{n} = G^{(0)} and ⋂_{n∈ℕ} (r, d) (W_{n}) = diag (G^{(0)}). Each W_{n} may be taken

to be a neighborhood of the unit space.

Moreover the distance function ρ may be taken to satisfy the following properties:

1. ρ (x, y) = ρ (x^{−1}, y^{−1}) for all x, y ∈ G.

2. ρ (x, r (x)) = ρ (x, d (x)) for all x ∈ G.

3. ρ (xy, r (x)) ≤ ρ (x, r (x)) + ρ (y, r (y)) for all (x, y) ∈ G^{(2)}.

4. ρ (x, y) ≤ ρ (x^{−1}y, d (x)) for all x, y ∈ G such that r (x) = r (y).

5. ρ (d (x) , d (y)) ≤ 2ρ (x, y) and ρ (r (x) , r (y)) ≤ 2ρ (x, y) for all x,y ∈ G.

Proof. (a) ⇒ (b). Let us assume that G is a metrizable locally transitive topological groupoid. Then G is paracompact topological groupoid. According to [10, p. 361-362], for each neighborhood W of G^{(0)}, there is a neighborhood W′ of G^{(0)} such that W′W′ ⊂ W. Then the family W of symmetric neighborhoods of the unit space is a G-uniformity. By Proposition 3.8, the topology τ_{W} induced by the G-uniformity W coincides with the topology of G. Applying [7, Metrization Theorem 13, p. 186] G is pseudometrizable if and only if its uniformity has a countable base. Since a base for the uniform structure U_{W} induced the topology τ_{W} is {U_{W} }_{W ∈W}, where

U_{W} = {(x, y) ∈ G × G : y ∈ W xW }

there is a countable family {W′_{n}}_{n∈ℕ} such that each W′_{n} is a neighborhood of G^{(0)} and for each W ∈ W there is n ∈ ℕ such that U_{W′_{n}} ⊂ U_{W} or equivalently, W′_{n}xW′_{n} ⊂ W xW for all x ∈ G. In particular, for each W ∈ W there is n ∈ ℕ such that W′_{n}W′_{n} ⊂ W W. Since for each W ∈ W, there is W_{1} ∈ W such that W_{1}W_{1} ⊂ W and for W_{1} there is n_{1} ∈ ℕ such that W′_{n_{1}}W′_{n_{1}} ⊂ W_{1}W_{1}, it follows that in fact for each for each W ∈ W, there is n_{1} ∈ ℕ such that W′_{n_{1}} ⊂ W′_{n_{1}}W′_{n_{1}} ⊂ W. Thus {W′_{n}}_{n∈ℕ} is a fundamental system of neighborhoods of G^{(0)}. Since G is Hausdorff, for each x ∉ G^{(0)} there is a neighborhood V of r (x) such that x ∉ V. Furthermore x ∉ V ∪ (G \ G^{r(x)}) and V ∪ (G \ G^{r(x)}) is a neighborhood of G^{(0)}. Thus ⋂_{W ∈W} W = G^{(0)} and therefore ⋂_{n∈ℕ} W′_{n} = G^{(0)}. Let u, v ∈ G^{(0)}

be such that u ≠ v. Since G is Hausdorff, G_{v}^{u} is closed and G \ G_{v}^{u} is a neighborhood of

G^{(0)}. Hence ⋂_{W ∈W} (r, d) (W) = diag (G^{(0)}). Consequently, ⋂_{n∈ℕ} (r, d) (W′_{n}) = diag (G^{(0)}).

(b) => (c) Since G is a paracompact topological groupoid, [10, p. 361-362], for each neighborhood W of G^{(0)}, there is a neighborhood W′ of G^{(0)} such that W′W′ ⊂ W.

(c) => (d) Let {W_{n}}_{n∈ℕ} be a fundamental system of neighborhoods of G^{(0)} such that

⋂_{n∈ℕ} W_{n} = G^{(0)} and ⋂_{n∈ℕ} (r, d) (W_{n}) = diag (G^{(0)}). Replacing W_{n} with W_{n} ∩ W_{n}^{−1}, we may

assume that W_{n} = W_{n}^{−1} for all n. Let W′_{0} = W_{0}. Inductively we construct a G-uniformity {W′_{n}}_{n∈ℕ} consisting in neighborhoods of G^{(0)}. Suppose a symmetric neighborhood W′_{n} of G^{(0)} has already been built. Let W′′ be a symmetric neighborhood of G^{(0)} such that W′′W′′ ⊂ W′_{n}. Let W′_{n+1}be a neighborhood of G^{(0)} such that W′_{n+1} ⊂ W′′ ∩ W_{n+1}. Thus {W′_{n}}_{n∈ℕ} is a G-uniformity. Moreover {W′_{n}}_{n∈ℕ} is a fundamental system of neighborhoods of G^{(0)}. Therefore it is compatible with the topology of the fibres and ⋂_{n∈ℕ} W′_{n} = G^{(0)} as

well as ⋂_{n∈ℕ} (r, d) (W′_{n}) = diag (G^{(0)}).

(d) => (a). Let W = {W_{n}}_{n∈ℕ} be countable G-uniformity compatible with the topology of the fibres such that ⋂_{n∈ℕ} W_{n} = G^{(0)} and ⋂_{n∈ℕ} (r, d) (W_{n}) = diag (G^{(0)}). Let

I = \{\frac{1}{2^{n}} , n ∈ ℕ\}. Let W_{n_{0}} ∈ W be such that W_{n_{0}}W_{n_{0}} ⊂ W_{0}. Let W′_{0} = W_{n_{0}}. Inductively we construct an I-indexed family {W′_{i}}_{i∈I}. Suppose that W′_{1/2^{n}} ∈ W has already been built. Since W is a G-uniformity, there is a W_{m_{n}} ∈ W such that W_{m_{n}}W_{m_{n}} ⊂ W′_{1/2^{n}}. Let W′_{1/2^{n+1}} ∈ W be such that W′_{1/2^{n+1}} ⊂ W_{m_{n}} ∩ W_{n+1}. Thus we obtain an I-indexed family W′ = {W′_{i}}_{i∈I} as in Theorem 2.5 that in addition satisfies G^{(0)} = ⋂_{i∈I} W′_{i} and

⋂_{i∈I} (r, d) (W′_{i}) = diag (G^{(0)}). Moreover W′ = {W′_{i}}_{i∈I} is compatible with the topology

of the fibres. Thus for every x ∈ G and every W′_{i} ∈ W′ there is W′_{i_{x}} ∈ W′ such that xW′_{i}x^{−1} ⊂ W′_{i}. Let f_{reg} be the function associated in Proposition 2.6 to f = f_{G^{(0)},W′}, where f = f_{G^{(0)},W′} is the function constructed in Theorem 2.5. For all x,y ∈ G, let us define

ρ (x, y) := \frac{1}{2} inf \{f_{reg} (x^{−1}sy) + f_{reg} (s) : s ∈ G_{r(y)}^{r(x)}\},

if G_{r(y)}^{r(x)} ≠ ∅ and ρ (x, y) := 1 otherwise. Let us note that G_{r(x)}^{r(y)} ≠ ∅ if and only if G_{r(y)}^{r(x)} ≠ ∅ and

ρ (x, y) = \frac{1}{2} inf \{f_{reg} (x^{−1}sy) + f_{reg} (s) : s ∈ G_{r(y)}^{r(x)}\}
= \frac{1}{2} inf \{f_{reg} (y^{−1}s^{−1}x) + f_{reg} (s^{−1}) : s ∈ G_{r(y)}^{r(x)}\}
= \frac{1}{2} inf \{f_{reg} (y^{−1}tx) + f_{reg} (t) : t ∈ G_{r(x)}^{r(y)}\}
= ρ (y, x) .

Thus ρ (x, y) = ρ (y, x).

Let us prove that if r (x) = r (y), then ρ (x, y) ≤ \frac{1}{2}f_{reg} (x^{−1}y). Indeed,

ρ (x, y) ≤ \frac{1}{2} (f_{reg} (x^{−1}r (x) y) + f_{reg} (r (x))) = \frac{1}{2}f_{reg} (x^{−1}y).

If x = y, then ρ (x, y) ≤ \frac{1}{2}f_{reg} (x^{−1}y) = 0.

Let x, y, z ∈ G and let us prove that ρ (x, z) ≤ ρ (x, y) + ρ (y, z). If ρ (x, y) = 1 or ρ (y, z) = 1, then obviously, ρ (x, z) ≤ 1 ≤ ρ (x, y) + ρ (y, z). If ρ (x, y) < 1 and ρ (y, z) < 1, then for every ε > 0 there are s_{1} = s_{1} (ε) ∈ G_{r(y)}^{r(x)} and s_{2} = s_{2} (ε) ∈ G_{r(z)}^{r(y)} such that ρ (x, y) > \frac{1}{2}f_{reg} (x^{−1}s_{1}y) + \frac{1}{2}f_{reg} (s_{1}) − ε and ρ (y, z) > \frac{1}{2}f_{reg} (y^{−1}s_{2}x) + \frac{1}{2}f_{reg} (s_{2}) − ε. Furthermore

ρ (x, z) ≤ \frac{1}{2}f_{reg} (x^{−1}s_{1}s_{2}z) + \frac{1}{2}f_{reg} (s_{1}s_{2})
≤ \frac{1}{2}f_{reg} (x^{−1}s_{1}yy^{−1}s_{2}z) + \frac{1}{2}f_{reg} (s_{1}) + \frac{1}{2}f_{reg} (s_{2})
≤ \frac{1}{2}f_{reg} (x^{−1}s_{1}y) + \frac{1}{2}f_{reg} (y^{−1}s_{2}z) + \frac{1}{2}f_{reg} (s_{1}) + \frac{1}{2}f_{reg} (s_{2})
< ρ (x, y) + ρ (y, z) + 2ε.

Therefore ρ (x, z) ≤ ρ (x, y) + ρ (y, z).

Let us show that if ρ (x, y) = 0, then x = y. If ρ (x, y) = 0, for every n there is s_{n} ∈ G_{r(y)}^{r(x)} such that f_{reg} (s_{n}) < \frac{1}{2^{n}} and f_{reg} (x^{−1}s_{n}y) < \frac{1}{2^{n}}. Taking into account that f_{reg} (s_{n}) < \frac{1}{2^{n}} applying Proposition 2.6, it follows that s_{n} ∈ W′_{1/2^{n−4}} and consequently,

(r (x) , r (y)) = (r, d) (s_{n}) ∈ (r, d) (W′_{1/2^{n−4}}). Since ⋂_{n∈ℕ} (r, d) (W′_{1/2^{n}}) = diag (G^{(0)}),

it follows that r (x) = r (y). Moreover since f_{reg} (x^{−1}s_{n}y) < \frac{1}{2^{n}}, it follows that y ∈ W′_{1/2^{n−4}}xW′_{1/2^{n−4}} for all n ≥ 4. Let W′_{x,n} ∈ W′ be such that x^{−1}W′_{x,n}x ⊂ W′_{1/2^{n+1}} and W′_{x,n} ⊂ W′_{1/2^{n+1}}. We have y ∈ W′_{x,n}xW′_{x,n}. Consequently, y ∈ xx^{−1}W′_{x,n}xW′_{x,n} ⊂

xW′_{1/2^{n+1}}W′_{1/2^{n+1}} ⊂ xW′_{1/2^{n}}. Hence x^{−1}y ∈ ⋂_{n∈ℕ} W′_{1/2^{n}} = G^{(0)}. Thus x = y.

We have proved that ρ is a metric on G. Let us prove that the topology defined by ρ coincides with the topology induced by the G-uniformity W′ and consequently, with the topology of G. Let y ∈ B (x, \frac{1}{2^{n}} ), n ∈ ℕ, n ≥ 6. Then there is s ∈ G_{r(y)}^{r(x)} such that f_{reg} (s) < \frac{1}{2^{n−2}} and f_{reg} (x^{−1}sy) < \frac{1}{2^{n−2}} . By Proposition 2.6, it follows that s ∈ W′_{1/2^{n−6}} and x^{−1}sy ∈ W′_{1/2^{n−6}}. Therefore y ∈ W′_{1/2^{n−6}}xW′_{1/2^{n−6}} and B (x, \frac{1}{2^{n}} ) ⊂ W′_{1/2^{n−6}}xW′_{1/2^{n−6}}. On the other hand for every n and x, if y ∈ W′_{1/2^{n}}xW′_{1/2^{n}}, then there are s, t ∈ W′_{1/2^{n}} such that y = sxt. Hence f_{reg} (x^{−1}s^{−1}y) = f_{reg} (x^{−1}xt) = f_{reg} (t) < \frac{1}{2^{n−1}} . Also ρ (x, y) ≤ \frac{1}{2} (f_{reg} (x^{−1}s^{−1}y) + f_{reg} (s^{−1})) = \frac{1}{2} (f_{reg} (t) + f_{reg} (s)) < \frac{1}{2^{n−3}} . Therefore W′_{1/2^{n}}xW′_{1/2^{n}} ⊂ B (x, \frac{1}{2^{n−3}} ).

Let us prove that ρ (x, y) = ρ (x^{−1}, y^{−1}) for all x,y ∈ G. We have G_{r(y)}^{r(x)} = ∅ if

and only if G_{r(y^{−1})}^{r(x^{−1})} = ∅. Thus if G_{r(y)}^{r(x)} = ∅, then ρ (x, y) = 1 = ρ (x^{−1}, y^{−1}). Let us

assume that G_{r(y)}^{r(x)} ≠ ∅. Then for every ε > 0 there is s_{ε} ∈ G_{r(y)}^{r(x)} such that ρ (x, y) > \frac{1}{2}f_{reg} (x^{−1}s_{ε}y) + \frac{1}{2}f_{reg} (s_{ε}) − ε. Let t = x^{−1}s_{ε}y. Then ρ (x^{−1}, y^{−1}) ≤ \frac{1}{2}f_{reg} (xty^{−1}) + \frac{1}{2}f_{reg} (t) = \frac{1}{2}f_{reg} (s_{ε}) + \frac{1}{2}f_{reg} (x^{−1}s_{ε}y) ≤ ρ (x, y) + ε. Similarly, ρ (x, y) ≤ ρ (x^{−1}, y^{−1}) + ε. Hence ρ (x, y) = ρ (x^{−1}, y^{−1}).

Let us show that ρ (x, r (x)) = ρ (x^{−1}, d (x)) = ρ (x, d (x)) = \frac{1}{2}f_{reg} (x) for all x ∈ G. We have ρ (x, r (x)) ≤ \frac{1}{2}f_{reg} (x^{−1}r (x)) = \frac{1}{2}f_{reg} (x). For all s ∈ G_{r(x)}^{r(x)} we have

\frac{1}{2}f_{reg} (x) = \frac{1}{2}f_{reg} (x^{−1}) = \frac{1}{2}f_{reg} (x^{−1}sr (x) s^{−1}) ≤ \frac{1}{2}f_{reg} (x^{−1}sr (x)) + \frac{1}{2}f_{reg}^{1/2} (s^{−1}). Thus ρ (x, r (x)) = \frac{1}{2}f_{reg} (x).

Also ρ (x^{−1}, d (x)) = \frac{1}{2}f_{reg} (x^{−1}) = \frac{1}{2}f_{reg} (x) = ρ (x, r (x)). Moreover ρ (x, d (x)) = ρ (x^{−1}, d (x)) for all x ∈ G.

For all (x, y) ∈ G^{(0)} we have ρ (xy, r (x)) = \frac{1}{2}f_{reg} (xy) ≤ \frac{1}{2}f_{reg} (x) + \frac{1}{2}f_{reg} (y) = ρ (x, r (x)) + ρ (y, r (y)).

If r (x) = r (y), then ρ (x, y) ≤ \frac{1}{2}f_{reg}^{1/2} (x^{−1}y) = ρ (x^{−1}y, d (x)) = ρ (y^{−1}x, d (y)).

Let us prove that ρ (d (x) , d (y)) ≤ 2ρ (x, y) and ρ (r (x) , r (y)) ≤ 2ρ (x, y) for all x,y ∈ G. Obviously, if G_{r(y)}^{r(x)} = ∅, then ρ (d (x) , d (y)) = ρ (r (x) , r (y)) = ρ (x, y) = 1. If

G_{r(y)}^{r(x)} ≠ ∅, then for every ε > 0 there is s_{ε} ∈ G_{r(y)}^{r(x)} such that ρ (x, y) > \frac{1}{2}f_{reg} (x^{−1}s_{ε}y) + \frac{1}{2}f_{reg} (s_{ε})−ε. Let t = x^{−1}s_{ε}y. Then ρ (d (x) , d (y)) ≤ \frac{1}{2}f_{reg} (t)+ \frac{1}{2}f_{reg} (t) = f_{reg} (x^{−1}s_{ε}y) ≤ f_{reg} (x^{−1}s_{ε}y) + f_{reg} (s_{ε}) < 2ρ (x, y) + 2ε. Hence ρ (d (x) , d (y)) ≤ 2ρ (x, y). We also have

ρ (r (x) , r (y)) = ρ (d (x^{−1}) , d (y^{−1})) ≤ ρ (x^{−1}, y^{−1}) = ρ (x, y)

for all x,y ∈ G.

References

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[3] M. Buneci, Haar systems and homomorphism on groupoids, Operator algebras and mathematical physics (Constanta, 2001), 35–49, Theta, Bucharest, 2003.

994  MĂDĂLINA ROXANA BUNECI

[4] M. Buneci, Convolution algebras for topological groupoids with locally compact fibres, Opuscula Mathematica 31 (2011), 159-172.

[5] M. Buneci, Various notions of amenability for not necessarily locally compact groupoids, Surveys in Mathematics and its Applications, 9 (2014), 55-78.

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[8] D. Mitrea, I. Mitrea, M. Mitrea and S. Monniaux, Groupoid metrization theory. With applications to analysis on quasi-metric spaces and functional analysis, Applied and Numerical Harmonic Analysis. New York, NY: Birkhäuser Springer, 2013.

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University Constantin Brâncuşi of Târgu-Jiu Calea Eroilor No.30, 210135 Târgu-Jiu, România.

Email: mbuneci@yahoo.com, ada@utgjiu.ro
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