Let X be a separable Hausdorff space such that for every $x \in X$ there exists an open neighborhood $U$ of $x$ such that $U$ is homeomorphic to an open subset of $\mathbb{R}^n$.
Show that:
(i) $X$ is locally compact
(ii)there exists a countable compact cover of $X$.
My attempt: (i) Suppose $x \in X$ and U to be an open neighborhood of $x$. Then there exists an open subset $V$ of $\mathbb{R}^n$ and a homeomorphism $\phi: U \rightarrow V$. $V$ is an open neighborhood of $\phi(x)$ and we can find an open ball with radius $r$, $B_r(\phi(x))$ such that $B_r(\phi(x)) \subseteq V$. Now,the closed ball with radius $r/2 $ $\overline{B_{\frac{r}{2}}(\phi(x))}$ is a compact neighborhood of $\phi(x)$ and since $\phi$ is a homeomorphism, $\phi^{-1}(\overline{B_{\frac{r}{2}}(\phi(x))})$ is a compact neighborhood of $x$ in $X$. Thus, X is locally compact.
(ii): Approach 1: I know that if $X$ is second countable, then any open cover of X, has an open countable subcover. But $X$ isn't a second countable space. So this could be the wrong approach. (Question: Is there a way to show that $X$ is second countable?)
Approach 2: If $X$ is separable, there exists a countable dense subset $D$ such that $\overline{D}=X$. Since $X$ is locally compact, we can choose a compact neighborhood $C_d$ for $d \in D$. If $X \subseteq \cup_{d \in D} C_d$, then we are finished. That's how far I got.
Is there a way to prove (ii) (maybe with a different approach)?
Searching the $\pi$-Base can help with these sorts of questions:
All locally euclidian spaces are locally compact: https://topology.pi-base.org/theorems/T000332/
In this case the $\pi$-Base doesn't know if such spaces are $\sigma$-compact or second countable.
From https://topology.pi-base.org/spaces?q=%24T_2%24%2BLocally+%24n%24-Euclidean%2B%7E%24%5Csigma%24-compact we see that the separable condition will be necessary.
From https://topology.pi-base.org/spaces?q=Separable%2BLocally+%24n%24-Euclidean%2B%7E%24%5Csigma%24-compact we see the Hausdorff condition would be necessary.
From https://topology.pi-base.org/spaces?q=%24T_2%24%2BSeparable%2B%7E%24%5Csigma%24-compact we see that the locally euclidian property would be necessary. From https://topology.pi-base.org/spaces?q=Separable%2B%24T_2%24%2BLocally+compact%2B%7ESecond+Countable we see that locally compact is an insufficient replacement.
One counterexample is described in the link from Chad K's comment. I was unable to find a peer-reviewed verison of the Gauld paper cited, but in 1990 Mary Ellen Rudin published a ZFC example which was $T_4$, separable, locally Euclidean, but not metrizable. Note that in this context, $\sigma$-compact, second countable, and metrizable are all equivalent. (I haven't worked through the details, but should note that she explicitly asserted that this space is both normal and Hausdorff.)