First-countable implies compactly generated

From Topospaces

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., first-countable space) must also satisfy the second topological space property (i.e., compactly generated space)
View all topological space property implications | View all topological space property non-implications
Get more facts about first-countable space|Get more facts about compactly generated space

Statement

Property-theoretic statement

The property of topological spaces of being first-countable is stronger than the property of being compactly generated.

Verbal statement

Any first-countable space is compactly generated.

Proof

Given: A first-countable space

To prove: There exists a collection of compact subsets of , such that is open if and only if is open in for every

Construction of the collection of compact subsets

We consider compact subsets of the following form: take a sequence that converges to a point (note that a sequence may converge to more than one point, we just make sure there is at least one point of convergence). Now define the compact set corresponding to this sequence as .

Let's see why is compact. First, observe that by construction, any open set of containing must contain all but finitely many . Hence, if we have an open cover of , the member of the open cover containing must contain all but finitely many of the s. Picking one member of the open cover for each , we obtain a finite subcover.

Proof that this collection works

Given: A subset of such that is open in for every of the above form.

To prove: is open in

Proof: We want to show that for every , there exists an open subset of containing and inside .

Since is first-countable, we can find, for any point , a descending chain of open subsets containing , such that every open set containing contains one of the s. It thus suffices to show that one of the s is contained in .

Suppose not. Then, none of the s is contained in . Pick for each , and consider the sequence of s. Any open subset containing contains one of the s, so it contains all but finitely many of the s. Thus, the s converge to Consider the compact set formed by the s and . By assumption, is open in . But this would force to contain one or more of the s, a contradiction. This completes the proof.