nLab CW-complexes are paracompact Hausdorff spaces

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 and Proof

Proposition

Every CW-complex is a paracompactum, i.e., a paracompact Hausdorff space.

For one textbook explanation, see e.g. Fritsch-Piccinini 90, theorem 1.3.5. Below we give a somewhat more categorically minded proof, linking to relevant results elsewhere in the nLab.

Proof

Let X nX_n denote the n thn^{th} skeleton of XX. We argue by induction that each skeleton is a paracompactum. Vacuously X 1=X_{-1} = \emptyset is a paracompactum. Now suppose X n1X_{n-1} is a paracompactum, and suppose X nX_n is formed as an attachment space with attaching map f: iIS i n1X n1f: \sum_{i \in I} S_i^{n-1} \to X_{n-1}, so that

iIS i n1 h iID i n f (po) g X n1 k X n\array{ \sum_{i \in I} S_i^{n-1} & \stackrel{\;h\;}{\hookrightarrow} & \sum_{i \in I} D_i^{n} \\ \mathllap{f} \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \mathrlap{g} \\ X_{n-1} & \underset{\;k\;}{\hookrightarrow} & X_n }

is a pushout square. The embedding hh is a closed embedding, and so its pushout along ff is a closed embedding kk. Furthermore, the spheres S i n1S_i^{n-1} and disks D i nD_i^n are paracompacta since they are compact Hausdorff, and coproducts of paracompacta are again paracompacta, making hh a closed embedding of paracompact Hausdorff spaces. By the result on pushouts of closed embeddings of paracompacta, it now follows that X nX_n is a paracompactum.

Thus the CW-complex XX is a colimit of a sequence of closed embeddings

X 1X 0X 1X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \ldots

between paracompacta. It follows that this colimit is a paracompactum.

References

That every CW complexes is Hausdorff (in fact normal) may be folklore, a proof is spelled out in:

An early original article with the paracompactness statement is:

  • Hiroshi Miyazaki, The paracompactness of CW-complexes, Tohoku Math. J. (2) Volume 4, Number 3 (1952), 309-313. 1952 Euclid

Textbook account:

Last revised on November 21, 2021 at 07:24:03. See the history of this page for a list of all contributions to it.