nLab sigma-topological space



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



A σ\sigma-topological space is a set XX equipped with a σ\sigma-topology: a collection O(X)O(X) of subsets of XX, called open sets, which is closed under finite intersections and countable unions.

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

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



Let X= nαX nX = \prod_{n \to \alpha} X_n be an internal set, and let X\mathcal{F}_X be the collection of all subsets of XX that can be expressed as the union of at most countably many internal subsets of XX. Then (X, X)(X, \mathcal{F}_X) is a countably compact T 1 T_1 σ\sigma-topological space.


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

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

Last revised on February 4, 2024 at 18:33:45. See the history of this page for a list of all contributions to it.