I'm working through Milnor and Stasheff's Characteristic Classes and got stuck in chapter 5, p.66, where some (supposedly) easy facts about paracompact spaces are assembled. One of these is:
Theorem of Morita. If a regular topological space is the countable union of compact subsets, then it is paracompact.
What I can't seem to wrap my head around is the following corollary:
Corollary. The direct limit of a sequence $K_1 \subset K_2 \subset K_3 \subset \ldots $ of compact spaces is paracompact. ...
For it follows from [Whitehead, 1961, §18.4] that such a direct limit is regular. (The reader should have no difficulty in supplying a proof.)
The referenced paper is
- J.H.C. Whitehead, Manifolds with transverse fields in Euclidean space, Annals of Math. 73 (1961), pp.154-212, JSTOR link.
I get how $\varinjlim_{i}K_i = \bigcup\limits_{i}K_i$ but I have no clue how to prove regularity when the $K_i$ aren't even assumed to be Hausdorff. The reference takes the $K_i$ to be normal Hausdorff spaces, so I don't see how that helps.