intersection of locally compact Hausdorff topologies.

Question

Are there locally compact Hausdorff topologies $\mathcal T, \mathcal S$ on a set $X$, such that $\mathcal T\cap \mathcal S$ is a Hausdorff but not locally compact topology on $X$?

Answer 1

Let $\mathscr{E}$ be the usual topology on $\mathbb{Q}$, and let $\mathscr{T}$ be the topology obtained by isolating each point of $\mathbb{Q}$ except $0$. Clearly $\langle\mathbb{Q},\mathscr{T}\rangle$ is Hausdorff, but not locally compact at $0$. Let $U_0=\{0\}\cup\{2^{-n}:n\in\mathbb{N}\}$, and let $U_1=\{0\}\cup\{-2^{-n}:n\in\mathbb{N}\}$. For $i\in\{0,1\}$ let $\mathscr{T_i}$ be the topology on $\mathbb{Q}$ generated by the subbase $\mathscr{T}\cup\{U_i\}$; $\mathscr{T_i}$ is finer than $\mathscr{T}$ and therefore Hausdorff, and $U_i$ is a compact clopen nbhd of $0$ in $\langle\mathbb{Q},\mathscr{T_i}\rangle$, which is therefore locally compact. Finally, $\mathscr{T}=\mathscr{T_0}\cap\mathscr{T_1}$.