# A first countable Hausdorff space is compactly generated

I know that even a non-Hausdorff first countable space is compactly generated, but I assume that adding the property that the space is also Hausdorff, there is an easier proof. How would you prove that a first countable Hausdorff space is compactly generated? I assume using the fact that a compact subspace in a Hausdorff space is closed is to key to make the proof easier, but I don't see how.

I use the following definition for a compactly generated space: A space is compactly generated if (i) a subspace $A$ is closed in $X$ if and only if (ii) $A\cap C$ is closed in $C$ for all compact subspaces $C\subseteq X$.

To show that (i) $\Rightarrow$ (ii) is easy. Since $X$ is a Hausdorff space, $C$ is closed and the intersection $A\cap C$ is an intersection between two closed sets and hence closed in both $C$ and $X$.

Suppose that $A$ is not closed; then there is an $x\in(\operatorname{cl}A)\setminus A$. Since $X$ is first countable, there is a sequence $\langle x_n:n\in\Bbb N\rangle$ in $A$ converging to $x$. Let $C=\{x\}\cup\{x_n:n\in\Bbb N\}$; then $C$ is a compact subset of $X$, but $A\cap C=\{x_n:n\in\Bbb N\}$ is not closed in $X$.
First countability of $X$ is actually more than is needed: it suffices to assume that $X$ is sequential. If a subset of a sequential space is not closed, there is a sequence in it converging to a point not in it, which is precisely what we need here.
• So $X$ being Hausdorff does not add anything of interest to the proof? – Barbara Nov 22 '16 at 21:30
• @BrianMScott: Your second $X$ should be $x$. I can not edit it because the edit is less then 6 characters. – Barbara Nov 24 '16 at 17:24
• What is $C$ compact? Is it a one-point-compactification (Alexandroff compactification)? – Barbara Nov 24 '16 at 17:39
• @Gjermund: Thanks for catching the typo. $C$ is compact because it’s a convergent sequence together with its limit: any open set that contains $x$ automatically contains all but finitely many of the $x_n$. (And yes, that does make it in effect the one-point compactification of a countably infinite discrete space.) – Brian M. Scott Nov 24 '16 at 18:19