Are the following topological spaces locally compact? I am trying to determine whether the following spaces are locally compact:
a) the slotted plane
b) the radial plane
For part a) I am almost certain that it is not compact, but not sure how to go about showing this.
Please show any steps to prove this or likewise, any counterexample to disprove.
Thanks!

HINT: For (a) you can show that if $U$ is a non-empty open set, then $U$ contains a line segment $L$. $L$ is a closed, discrete set in the whole space, and it has cardinality $\mathfrak{c}$. And $\operatorname{cl}U$ is separable, so by Jones's lemma it is not normal. It Hausdorff, however, so it cannot be compact; why?
For (b) you can use the same basic idea. First show that $\Bbb Q^2$ is dense in the radial plane. Then show that every circle is a closed, discrete set in the radial plane. Finally, use the fact that $\mathfrak{c}$ is not the supremum of countably many smaller cardinals to show that if $U$ is an open nbhd of a point $p$, then there is an $n\in\Bbb N$ such that $U$ contains $\mathfrak{c}$ points of the circle of radius $2^{-n}$ centred at $p$. Then apply Jones's lemma and argue as before.
