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Convergence in exponentiable spaces

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Claudio Pisani

Exponentiable spaces are characterized in terms of convergence.
More precisely, we prove that a relation $R:{\cal U}X \rightharpoonup X$
between
ultrafilters and elements of a set $X$ is the convergence relation for a
quasi-locally-compact (that is, exponentiable)
topology on $X$ if and only if the following conditions are satisfied:

1. $id \subseteq R\circ\eta $

2.$R\circ {\cal U}R = R\circ\mu $

where $\eta : X \to {\cal U}X$ and $\mu : {\cal U}({\cal U}X) \to
{\cal U}X$
are the unit and the multiplication of the ultrafilter monad,
and ${\cal U} : \bi{Rel} \to \bi{Rel}$ extends the ultrafilter functor
${\cal U} : \bi{Set} \to \bi{Set}$ to the category of sets and
relations.
$({\cal U},\eta,\mu)$ fails to be a monad on $\bi{Rel}$ only because
$\eta$ is not a strict natural transformation. So, exponentiable spaces
are the lax (with respect to the unit law) algebras for a lax monad on
$\bi{Rel}$. Strict algebras are exponentiable and $T_1$ spaces.

Keywords: exponentiable spaces, quasi-local-compactness, convergence, ultrafilter
monad, lax monads and algebras, continuous lattices.

1991 MSC: Primary 54A20, 54D45; Secondary 18C15.

*Theory and Applications of Categories*, Vol. 5, 1999, No. 6, pp 148-162.

http://www.tac.mta.ca/tac/volumes/1999/n6/n6.dvi

http://www.tac.mta.ca/tac/volumes/1999/n6/n6.ps

http://www.tac.mta.ca/tac/volumes/1999/n6/n6.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1999/n6/n6.dvi

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