A topological space $X$ is called locally compact if every point $x \in X$ has a compact neighborhood. (It seems that this notion of "locally compact" is sometimes called weakly locally compact.)
I know that in a locally compact Hausdorff space all open, closed and locally closed subspaces are again locally compact. What is an example for an arbitrary subspace which is not locally compact?
Also, I am quite lost what happens if one drops the Hausdorff assumption. Do the above results still hold? If not, what are counterexamples?
Best Answer
In $\Bbb R$, endowed with its usual topology, $\Bbb Q$ is not locally compact. Actually, in $\Bbb Q$, if $V$ is any neighborhood of any point, then $V$ is not compact.