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Profinite topological spaces

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G. Bezhanishvili, D. Gabelaia, M. Jibladze, P. J. Morandi

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

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