first countable implies compactly generated


Suppose X is first countable, and AX has the property that, if C is any compact set in X, the set AC is closed in C. We want to show tht A is closed in X. Since X is first countable, this is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to showing that any sequence (xi) in A converging to x implies that xA. Let C={xii=1,2,}{x}.

Lemma 1.

C is compact.


Let {UjjJ} be a collectionMathworldPlanetmath of open sets covering C. So xUj for some j. Since Uj is open, there is a positive integer k such that xiUj for all ik. Now, each xiUd(i) for i=1,,k. So C is covered by Ud(1),,Ud(k), and Uj, showing that C is compact. ∎

In addition, as a subspaceMathworldPlanetmathPlanetmath of X, C is also first countable. By assumptionPlanetmathPlanetmath, AC is closed in C. Since xiAC for all i1, we see that xAC as well, since C is first countable. Hence xA, and A is closed in X. ∎

Title first countable implies compactly generated
Canonical name FirstCountableImpliesCompactlyGenerated
Date of creation 2013-03-22 19:09:35
Last modified on 2013-03-22 19:09:35
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 4
Author CWoo (3771)
Entry type Example
Classification msc 54E99