nLab
second-countable regular spaces are paracompact

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

Lemma

(Michael's theorem, Michael 53, theorem 1)

Let XX be a topological space such that

  1. XX is regular;

  2. every open cover of XX has a refinement by a union of a countable set of locally finite sets of open subsets (not necessarily covering).

Then XX is paracompact topological space.

Proposition

(second-countable regular spaces are paracompact)

Let XX be a topological space which is

  1. second-countable;

  2. regular.

Then XX is paracompact topological space.

Proof

Let {U iX} iI\{U_i \subset X\}_{i \in I} be an open cover. By Michael's theorem (lemma 1) it is sufficient that we find a refinement by a countable cover.

But second countability implies precisely that every open cover has a countable subcover:

Every open cover has a refinement by a cover consisting of base elements, and if there is only a countable set of these, then the resulting refinement necessarily contains at most this countable set of distinct open subsets.

References

Revised on August 9, 2017 04:58:41 by Urs Schreiber (46.183.103.8)