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CW-complexes are paracompact Hausdorff spaces

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Topology

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

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

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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} \downarrow & po & \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 is 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

An early original article with the statement is

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

Textbook accounts include

  • Rudolf Fritsch, Renzo Piccinini, Theorem 1.3.5 (p. 29 and following) of Cellular structures in topology, Cambridge University Press (1990)

Revised on June 22, 2017 14:23:37 by Eric Auld? (2607:f010:2e9:9:bd08:48c7:369d:ff7d)