Examples of paracompact spaces

Example 9.1.3. The following classes of topological spaces are paracompact:

• Every compact Hausdorff space is paracompact [Hatcher (2003), Proposition 1.18];
• Every direct limit, with the colimit topology, of an increasing sequence \(X_1 \subseteq X_2 \subseteq \dots \) of compact Hausdorff spaces is paracompact [Hatcher (2003), Proposition 1.19];
• Every CW-complex is paracompact [Hatcher (2003), Proposition 1.20].