# spaces for which every open set is a union of closed sets

In the context of generalized metrizability of spaces I'm interested in the property of a topological space that every open set in it is a union of closed sets. A google search led me to: http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2003;task=show_msg;msg=0048.0002 however, I do not have the mentioned book nor will it be likely I can get it any time soon. Specifically, it would appear the book calls such spaces $\pi_1$ spaces and it proves that a space is $\pi_1$ precisely when every principal a-ideal in the lattice of open sets is normal.

I would really like to understand this theorem. Unfortunately, I don't know what a-ideal means nor what normality for an ideal in that context is.

So, if anybody can contribute to deciphering the meaning of these concepts and/or point to an online available proof and/or supply a proof I will be very grateful.

I am also interested in the hierarchy of $\pi_i$ properties for spaces.

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## 1 Answer

$\newcommand{\ms}{\mathscr}$Vaidyanathaswamy’s property $\Pi_1$ is that every open set is a union of closed sets. His characterization theorem is:

Theorem. The topology defined by the lattice $\Gamma$ of open sets is a $\Pi_1$-topology if and only if every principal $\alpha$-ideal of $\Gamma$ is normal.

His $\alpha$-ideals are upper ideals, what I would call filters: if $\langle K,\lor,\land,1\rangle$ is a distributive lattice with maximum element $1$, a set $F\subseteq K$ is an $\alpha$-ideal iff $F$ is closed under $\land$ and $F={\uparrow\!F}$. A principal $\alpha$-ideal is what I’d call a principle filter: $\uparrow\!a$ for some $a\in K$. The definition of normality and the machinery behind the proof seem to be a bit involved, judging by what I can see at Google Books. However,

Miss S. Pankajam, Ideal Theory in Boolean Algebra and Its Application to Deductive Systems, Proceedings of the Indian Academy of Sciences, Volume 14, Number 6 (1941), 670-684

seems to cover most of what’s needed and possibly all. That journal is open-access, but the older issues aren’t (yet?) directly accessible. However, if you go here, you can download PDFs of individual pages; you want PDFs $00000727$ through $00000741$. This paper by Vaidyanathaswamy may also be helpful, as it deals with the topological setting. (It’s all sufficiently involved that I’ve not done much more than skim to be sure that these references are likely to be useful.)

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@Ross: Forgot to prefix the URL with http://, so it didn’t show up as a link. Thanks for catching it. – Brian M. Scott Nov 17 '12 at 4:24
Fabulous! Thanks Brian. – Ittay Weiss Nov 17 '12 at 4:49
@Ittay: My pleasure! – Brian M. Scott Nov 17 '12 at 4:49