nLab
fully normal spaces are equivalently paracompact

Contents

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

Let XX be a T 1T_1 topological space.

Assuming the axiom of choice then the following are equivalent:

  1. XX is a fully normal topological space;

  2. XX is a paracompact Hausdorff topological space.

(Stone 48)

Since metric spaces are fully normal it follows as a corollary that metric spaces are paracompact. Accordingly, this statement is now also known as Stone’s theorem.

Note that without a separation axiom such as T 1T_1, the result fails to hold. For example, any compact topological space is paracompact, and any fully normal topological space is normal, so any non-normal compact space is a paracompact space that’s not fully normal.

The Hausdorff condition from statement 2 cannot be dropped, again because there are compact T 1T_1 spaces that are not normal, such as an infinite set with the cofinite topology.

References

  • A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. Volume 54, Number 10 (1948), 977-982. (Euclid)

Last revised on June 16, 2021 at 04:07:44. See the history of this page for a list of all contributions to it.