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Given universal set $U$. Is there any name of the collection of subsets of $U$ (call them quasi-open) satisfying the following axioms:

i) $\emptyset$ and $U$ are quasi-open;

ii) finite intersections of quasi-open sets are quasi-open;

iii) countable unions of quasi-open sets are quasi-open?

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If you also allow countable intersections then you will get $\sigma$-algebra, but I guess this is not what you want. – user43326 Jul 4 '14 at 10:34
@user43326 No, there is no condition on complements. – PVAL Jul 4 '14 at 10:41
I would think this as a topological space with a given basis of the topology which is stable by countable unions. – Pietro Majer Jul 4 '14 at 10:59
up vote 8 down vote accepted

This type of structure has sometimes been called a "$\sigma$-topology", especially in connection with ultraproducts. For instance the invariant Loeb measure on an ultrafinite group can be thought of as a sort of Haar measure on a $\sigma$-topological group. See for instance Section 2 of this paper of Bergelson and Tao:

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Let's call a collection of subsets of $X$ containing the empty set, $X$ and closed under finite intersection and countable union a $\sigma$-topology. The most prominent example of a $\sigma$-topology is the collection of all cozero sets in a topological space. A zero set in a topological space $X$ is a set of the form $f^{-1}[0]$ for some continuous $f:X\rightarrow\mathbb{R}$, and the complement of a zero set is a cozero set. The collection of all cozero sets has a prominent position in general topology since the collection of cozero sets forms a basis for any completely regular space. As expected, notions such as the Stone-Cech compactification, and the Hewitt realcompactification can be defined in terms of ultrafilters on the lattice of zero sets. Furthermore, the notion of pseudocompactness can be characerized in terms of zero sets and cozero sets; a completely regular space is pseudocompact if and only if every countable cozero cover has a finite subcover. I can say a bit more about the importance of zero-sets but I will refer the reader to the old but still standard basic reference [2].

In the paper [1], the author defines a zero-set structure in a way that abstractly axiomatizes what is meant by a zero set. Of course, by taking complements, we can abstractly axiomatize what is meant by a cozero set as well. Working with zero-set structures instead of just $\sigma$-topologies is very much within the spirit of general topology since one usually considers spaces and other structures satisfying substantial separation axioms instead of simply topological spaces without any separation axioms. In other words, if one is at all concerned about $\sigma$-topologies and spaces that satisfy separation axioms (of at least complete regularity), then one should consider zero-set structures instead of simply $\sigma$-topologies.

A zero-set structure is defined in [1] to be a collection $\mathcal{Z}$ of subsets of some set $X$ such that

  1. For every pair of distinct points in $X$ there is a $Z\in\mathcal{Z}$ containing precisely one of these points.

  2. $\mathcal{Z}$ is a $\sigma$-topology on $X$.

  3. If $Z\in\mathcal{Z}$ then there are $Z_{n}\in\mathcal{Z}$ for all natural numbers $n$ such that $Z^{c}=\bigcup_{n\in\mathbb{N}}Z_{n}$.

  4. If $Z_{1},Z_{2}\in\mathcal{Z}$ and $Z_{1}\cap Z_{2}=\emptyset$, then there are $V_{1},V_{2}\in\mathcal{Z}$ with $Z_{1}\subseteq V_{1}^{c},Z_{2}\subseteq V_{2}^{c}$ and $V_{1}^{c}\cap V_{2}^{c}=\emptyset$.

Of course, if $X$ is a completely regular space, then the collection of all zero-sets in $X$ is a zero-set structure. More generally, if $Y$ is a completely regular space and $X\subseteq Y$, then $\{Z\cap X|Z\,\textrm{is a zero set in Y}\}$ is a zero set structure on $X$ and any zero-set structure is of this form.

Zero-sets structures can also be characterized as the zero sets in certain rings of functions.

If $\mathcal{A}$ is a ring of functions from some set $X$ to $\mathbb{R}$ that is closed under uniform convergence, separates points, and contains all constants, then $\{f^{-1}[0]|f\in\mathcal{Z}\}$ is a zero-set structure.

Conversely, if $\mathcal{Z}$ is a zero-set structure on a set $X$, then define $\mathcal{A}_{\mathcal{Z}}$ to be the collection of all functions $f:X\rightarrow\mathbb{Z}$ such that $f^{-1}[C]\in\mathcal{Z}$ for each closed set $C$. Then $\mathcal{A}_{\mathcal{Z}}$ is a ring of functions closed under uniform convergence, separates points, and contains all constants and $\mathcal{Z}=\{f^{-1}[0]|f\in\mathcal{A}_{\mathcal{Z}}\}$.

Zero-set structures can also be characterized in terms of proximity spaces. In fact, the notion of a zero-set structure lies somewhere between the notion of a completely regular space and a proximity space in terms of how much structure one has.

If $(X,\delta)$ is a proximity space, then a set of the form $f^{-1}[\{0\}]$ for some proximity map $f:(X,\delta)\rightarrow[0,1]$ is said to be a proximally zero set. If $(X,\delta)$ is a proximity space, then the collection of all proximally zero sets forms a zero-set structures. Furthermore, if $(X,\mathcal{Z})$ is a zero-set structure, then there is a proximity $\delta$ on $X$ such that $\mathcal{Z}=\{f^{-1}[\{0\}|f:(X,\delta)\rightarrow[0,1]\,\textrm{is a proximity map}\}$.

At last, in point-free topology, the structures analogous to $\sigma$-topologies are called $\sigma$-frames. A $\sigma$-frame is therefore defined to be a poset $L$ with a least and greatest element such that every finite subset of $L$ has a least upper bound, every countable subset has a least upper bound, and $x\wedge\bigvee_{n\in\omega}y_{n}=\bigvee_{n\in\omega}(x\wedge y_{n})$. One can define the notions of regularity, complete regularity, and normality not only for frames, but also for $\sigma$-frames. Furthermore, the notion of a $\sigma$-frame is quite nice since every regular $\sigma$-frame is normal and completely regular as a $\sigma$-frame.

I think I have said enough, but I will end with a question that seems to be in the affirmative but which I have not researched thoroughly.

$\mathbf{Question}$ Are the regular $\sigma$-frames precisely the quotients $\mathcal{Z}/\simeq$ where $\mathcal{Z}$ is some zero-set structure and $\simeq$ is some congruence on the lattice $\mathcal{Z}$ preserving countable joins?

[1] Rings of functions determined by zero-sets. Hugh Gordon. Pacific J. Math. Volume 36, Number 1 (1971), 133-157.

[2] Gillman, Leonard, and Meyer Jerison. Rings of Continuous Functions. Princeton, NJ: Van Nostrand, 1960.

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In lattice theory, this would be called something like an ω-complete set-lattice. The algebraic analogue would be an ω-complete distributive lattice, and I'd have to check but I think the usual Stone representation preserves chain-completeness and more specifically ω-completeness.

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