nLab sigma-topological space

 Idea

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 finite intersections and countable unions.

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.

 Related concepts

• topological space
• sigma-frame
• sigma-locale
• sigma-algebra
• zero-set structure
• countably compact space
• first countable space

References

• Fedor Petrov, “countable” topology, MathOverflow (web)

• Vitaly Bergelson?, Terence Tao, Multiple recurrence in quasirandom groups. (arXiv:1211.6372)