nLab second-countable regular spaces are paracompact




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

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(Michael's theorem, Michael 53, theorem 1)

Let XX be a topological space such that

  1. XX is regular;

  2. every open cover of XX has a refinement by a union of a countable set of locally finite sets of open subsets (not necessarily covering).

Then XX is paracompact topological space.

Note that while in Michael’s paper he assumes that paracompact spaces are Hausdorff, the assumption is not necessary, and the proof goes through for regular non-Hausdorff spaces. See Kelley, p. 156.


(second-countable regular spaces are paracompact)

Let XX be a topological space which is

  1. second-countable;

  2. regular.

Then XX is paracompact topological space.


Let {U iX} iI\{U_i \subset X\}_{i \in I} be an open cover. By Michael's theorem (lemma ) it is sufficient that we find a refinement by a countable cover.

But second countability implies precisely that every open cover has a countable subcover:

Every open cover has a refinement by a cover consisting of base elements, and if there is only a countable set of these, then the resulting refinement necessarily contains at most this countable set of distinct open subsets.


Last revised on May 19, 2020 at 05:25:19. See the history of this page for a list of all contributions to it.