nLab sigma-topological space

Context

Topology

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

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

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

 Idea

In variation of the standard definition of topological spaces, 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

  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)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.

 Properties

Theorem

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.

References

Last revised on June 24, 2024 at 10:17:28. See the history of this page for a list of all contributions to it.