It is well known that profinite $T_0$-spaces are
exactly the spectral spaces. We generalize this result to the category
of all topological spaces by showing that the following conditions are
equivalent:
(1) $(X,\tau)$ is a profinite topological space.
(2) The $T_0$-reflection of $(X,\tau)$ is a profinite $T_0$-space.
(3) $(X,\tau)$ is a quasi spectral space.
(4) $(X,\tau)$ admits a stronger Stone topology $\pi$ such that $(X,
\tau,\pi)$ is a bitopological quasi spectral space
Keywords: Profinite space, spectral space, stably compact space, bitopological space, ordered topological space, Priestley space
2010 MSC: 18B30, 18A30, 54E55, 54F05, 06E15
Theory and Applications of Categories, Vol. 30, 2015, No. 53, pp 1841-1863.
Published 2015-12-03.
http://www.tac.mta.ca/tac/volumes/30/53/30-53.pdf
Revised 2016-03-07 (only to correct issue number). Original version at
http://www.tac.mta.ca/tac/volumes/30/53/30-53a.pdf