nLab CW-complexes are paracompact Hausdorff spaces

Statement and Proof

Proposition

Every CW-complex is 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 $X_n$ denote the $n^{th}$ skeleton of $X$. We argue by induction that each skeleton is a paracompact Hausdorff space. Vacuously $X_{-1} = \emptyset$ is a paracompact Hausdorff space. Now suppose $X_{n-1}$ is a paracompact Hausdorff space, and suppose $X_n$ is formed as an attachment space with attaching map $f: \sum_{i \in I} S_i^{n-1} \to X_{n-1}$, so that

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

is a pushout square. The embedding $h$ is a closed embedding, and so its pushout along $f$ is a closed embedding $k$. Furthermore, the spheres $S_i^{n-1}$ and disks $D_i^n$ are paracompact Hausdorff spaes since they are compact Hausdorff, and paracompact Hausdorff spaces are closed under coproducts, making $h$ a closed embedding of paracompact Hausdorff spaces. By the result on pushouts of closed embeddings of paracompact Hausdorff spaces, it now follows that $X_n$ is a paracompact Hausdorff space.

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

$$X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \ldots$$

between paracompact Hausdorff spaces. It follows that this colimit is a paracompact Hausdorff space.

Related statements

• Hausdorff implies sober

• maps from compact spaces to Hausdorff spaces are closed and proper

• closed subsets of compact spaces are compact

• compact subspaces of Hausdorff spaces are closed

• closed subspaces of compact Hausdorff spaces are equivalently compact subspaces

References

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

• Allen Hatcher, Prop. A.3 of Algebraic Topology, Cambridge University Press 2002 (ISBN:9780521795401, webpage)

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:

• Rudolf Fritsch, Renzo Piccinini, Theorem 1.3.5 (p. 29 and following) of: Cellular structures in topology, Cambridge University Press (1990) (doi:10.1017/CBO9780511983948, pdf)