nLab
Hausdorff implies sober

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

Statement

Proposition

Assuming excluded middle, then:

Every Hausdorff topological space is a sober topological space.

More specifically, in a Hausdorff topological space the irreducible closed subspaces are precisely the singleton subspaces.

Proof

The second statement clearly implies the first. To see the second statement, suppose that FF is an irreducible closed subspace which contained two distinct points xyx \neq y. Then by the Hausdorff property there are disjoint neighbourhoods U x,U yU_x, U_y, and hence it would follow that the relative complements F\U xF \backslash U_x and F\U yF \backslash U_y were distinct proper closed subsets of FF with

F=(F\U x)(F\U y) F = (F \backslash U_x) \cup (F \backslash U_y)

in contradiction to the assumption that FF is irreducible.

This proves by contradiction that every irreducible closed subset is a singleton. Conversely, generally the topological closure of every singleton is irreducible closed.

References

Revised on April 14, 2017 13:27:55 by Urs Schreiber (46.183.103.8)