locally compact Hausdorff space which is not second-countable

Question

I'm trying to find an example of a space that is Hausdorff and locally compact that is not second countable, but I'm stuck. I search an example on the book Counterexamples in Topology, but I can't find anything.
Thank you for any help.

Answer 1

Let $Y$ be an uncountable set, let $p$ be a point not in $Y$, and let $X=\{p\}\cup Y$. Let

$$\mathscr{B}=\{\{x\}:x\in Y\}\cup\{X\setminus F:F\text{ is a finite subset of }Y\}\;;$$

then $\mathscr{B}$ is a base for a Hausdorff, locally compact, compact topology $\tau$ on $X$ that is not second countable. It isn’t even first countable: $p$ has no countable local base. $\langle X,\tau\rangle$ is the one-point compactification of the discrete space $Y$. (Of course $Y$ itself is also an example, but it’s merely locally compact; $X$ is also compact.)

You can make examples with even nicer properties. For instance, $Y\times[0,1]$ is Hausdorff, locally compact, and locally connected but not second countable. The closed long ray has all of those properties and is in addition connected, path connected, and locally path connected. Its one-point compactification loses path connectedness but gains compactness.

Answer 2

Hint: Discrete spaces are locally compact.