Is a locally compact hereditarily Lindelof Hausdorff space first countable?

Question

Is a locally compact hereditarily Lindelof Hausdorff space first countable?

I was recently told that it is but I can't find any reference to what I would have thought would be a standard fact if it is correct.

Answer 1

Let $X$ be a locally compact hereditarily Lindelöf Hausdorff space and let $x\in X$. Then $X\setminus\{x\}$ is covered by sets of the form $X\setminus K$ where $K$ is a compact neighborhood of $x$. Since $X\setminus\{x\}$ is Lindelöf, it is in fact covered by countably many such sets $X\setminus K_n$, and we may assume the $K_n$ are nested. Thus we have a nested sequence of compact neighborhoods $K_n$ of $x$ such that $\bigcap K_n=\{x\}$. I claim that these are in fact a neighborhood base at $x$.

To prove this, suppose $U$ is a neighborhood of $x$ that does not contain any $K_n$. Pick a point $x_n\in K_n\setminus U$ for each $n$. Then $x_n\in K_0$ for all $n$, so by compactness the sequence $(x_n)$ accumulates somewhere in $K_0$. However, since the sequence is eventually in each $K_n$, any accumulation point must be in each $K_n$. Since $\bigcap K_n=\{x\}$, this means the accumulation point can only be $x$. But then since $U$ is a neighborhood of $x$, infinitely many of the $x_n$ must be in $U$. This is a contradiction, since $x_n\not\in U$ for all $n$.

Answer 2

It’s true. Since first countability is a local property, it suffices to show that a compact hereditarily Lindelöf Hausdorff space is first countable. If $X$ is a Hausdorff space, $\psi(X)\le hL(X)$, where $\psi(X)$ is the pseudocharacter of $X$ and $hL(X)$ is the hereditary Lindelöf degree of $X$. If $X$ is hereditarily Lindelöf, then $hL(X)=\omega$, so $\psi(X)=\omega$. And if $X$ is compact and Hausdorff, then $\psi(X)=\chi(X)$, the character of $X$, so in this setting we get $\chi(X)=\omega$, i.e., $X$ is first countable. Both of these results are noted without proof in R. Hodel, Cardinal Function I, in The Handbook of Set-Theoretic Topology, K. Kunen and J.E. Vaughan, eds.: Theorem $\bf{(3.8)}$$(f)$, and Theorem $\bf{7.1}$. There’s a proof of the former in I. Juhász, Cardinal Functions in Topology, Mathematical Centre Tracts 34, Theorem $\bf2.17$.