Equivalence of various notions of Compactness in Locally Compact spaces

Question

If $X$ is locally compact Hausdorff, then the following are equivalent -

1. $X$ is Lindelof
2. $X$ is $\sigma$-compact
3. $X$ is hemicompact
4. There exists a sequence $(A_n)$ of compact subsets of $X$ such that $A_n\subseteq A_{n+1}^\circ$ and $X = \bigcup\limits_{n=1}^\infty A_n$
5. $X$ is compact, or $\chi(\infty,X^*) = \mathbb N_0$
   Here, $X^* = X\cup\{\infty\}$ is the one-point compactification of $X$, and $\chi(x,X)$ is the character of the point $x$ in $X$.

Character of a Point: $\chi(x,X)$ is the smallest cardinal number of the form $|\mathscr B(x)|$, where $\mathscr B(x)$ is a neighbourhood base of $x$ in $X$.

I was able to prove $4\implies 3$, $3\implies 2$, and $2\implies 1$. I'm having trouble with the implications $5\implies 4$ and $1\implies 5$, mainly due to myy unfamiliarity with the concept of the Character of a Point. Any help will be appreciated!

Answer 1

$\newcommand{\cl}{\operatorname{cl}}$For $(5)\implies(4)$ there’s nothing to prove if $X$ is compact, so assume that $X^*$ is first countable at the point at infinity (i.e., that $\chi(\infty,X^*)=\omega$). Then $\infty$ has a local base $\mathscr{B}=\{B_n:n\in\omega\}$ such that $\cl_{X^*} B_{n+1}\subseteq B_n$ for each $n\in\omega$. Now let $A_n=X\setminus B_n$ for $n\in\omega$.

For $(1)\implies(5)$ there is nothing to prove if $X$ is compact, so assume that it is not. Use the fact that $X$ is locally compact and Lindelöf to get a countable open cover $\mathscr{V}=\{V_n:n\in\omega\}$ of $X$ such that $\cl_X V_n$ is compact for each $n\in\omega$. For each $n\in\omega$ let $B_n=X^*\setminus\cl_X V_n$; then $\mathscr{B}=\{B_n:n\in\omega\}$ is a countable family of open nbhds of $\infty$ such that $\bigcap_{n\in\omega}B_n=\{\infty\}$. In fact, it’s even true that $\bigcap_{n\in\omega}\cl B_n=\{\infty\}$.

Now let $U$ be any open nbhd of $\infty$. Then $$\bigcap_{n\in\omega}\big((\cl_{X^*} B_n)\setminus U\big)=\left(\bigcap_{n\in\omega}\cl_{X^*} B_n\right)\setminus U=\varnothing\,,$$ and the sets $(\cl_{X^*} B_n)\setminus U$ are closed in the compact Hausdorff space $X^*$, so there is a finite $F\subseteq\omega$ such that $$\bigcap_{n\in F}\big((\cl_{X^*} B_n)\setminus U\big)=\varnothing\,,$$ i.e., such that $\bigcap_{n\in F}\cl_{X^*} B_n\subseteq U$. Clearly, then, $\bigcap_{n\in F}B_n\subseteq U$, and you should now be able to identify a countable local base at $\infty$.