nLab sigma-topological space

 Idea

In variation of the standard definition of topological spaces, a $\sigma$-topological space is a set $X$ equipped with a $\sigma$-topology: a collection $O(X)$ of subsets of $X$, called open sets, which is closed under

1. finite$\;$intersections,

2. countable$\;$unions.

(the difference being that for an ordinary topology one requires closure under arbitrary unions, see there).

The open sets $O(X)$ of a $\sigma$-topological space form a $\sigma$-frame.

An example of a $\sigma$-topological space is a $\sigma$-algebra, whose collection $O(X)$ is also closed under complements and countable intersections. Another example of a $\sigma$-topological space is a zero-set structure.

 Properties

Theorem

Let $X = \prod_{n \to \alpha} X_n$ be an internal set, and let $\mathcal{F}_X$ be the collection of all subsets of $X$ that can be expressed as the union of at most countably many internal subsets of $X$. Then $(X, \mathcal{F}_X)$ is a countably compact $T_1$ $\sigma$-topological space.

In constructive mathematics

In constructive mathematics, there may be non-trivial $\sigma$-subframes of the frame of truth values. As a result, in analogy with the case for admissible Archimedean ordered fields, let us say that given a $\sigma$-subframe of the frame of truth values $\Sigma \subseteq \Omega$, a $\sigma$-topological space $(X, O(x))$ is $\Sigma$-admissible if and only if there is a function $(-) \in_\Sigma (-):X \times O(x) \to \Sigma$ such that $(x \in_\Sigma A) = \top$ if and only if $x \in A$.

Assuming the limited principle of omniscience, the boolean domain is the initial $\sigma$-frame and simultaneously a $\sigma$-subframe of the frame of truth values. In addition, by definition of boolean-admissible $\sigma$-topological spaces, the open subsets are decidable subsets of the $\sigma$-topological spaces. As a result, the boolean-admissible $\sigma$-topological spaces are precisely the sigma-topological spaces found in classical mathematics.

 Related concepts

• topological space

• sigma-frame

• sigma-locale

• sigma-algebra

• zero-set structure

• countably compact space

• first countable space

• sober sigma-topological space

References

• Fedor Petrov, “countable” topology, MathOverflow (2014) [MO:q/173255]

• Balazs Szegedy, Gowers norms, regularization and limits of functions on abelian groups [arXiv:1010.6211]

• Vitaly Bergelson, Terence Tao, Multiple recurrence in quasirandom groups, Geom. Funct. Anal. 24 (2014) 1–48 [arXiv:1211.6372, doi:10.1007/s00039-014-0252-0]