Is locally compact space Hausdorff? In this proof of the statement that proper maps to locally compact spaces are closed, the fact that compact subspaces of Hausdorff spaces are closed is used. However, is it true that locally compact spaces are Hausdorff?
 A: To clarify the definitions, a topological space $(X,\tau)$ is compact if every collection of open sets (i.e. elements of $\tau$) whose union is $X$ has a finite subcollection whose union is still $X$. The space is locally compact if for all $x\in X$ there exists an open set $U\in \tau$ and a subset $K\subseteq X$ such that $\{x\}\subseteq U\subseteq K$ and the subspace topology $(K,\tau|_K)$ is compact. A topological space $(X,\tau)$ is Hausdorff if for all $x,y\in X$ such that $x\not=y$, there exist open sets $U,V\in \tau$ such that $U\cap V=\varnothing$ and $x\in U$ and $y\in V$.
The following example of a locally compact space which is not Hausdorff was suggested in the comments.
Claim. Let $X$ be any set that contains at least $2$ distinct elements, and let $\tau=\{\varnothing,X\}$ be the topology whose only open sets are the empty set and $X$ itself. Then $(X,\tau)$ is locally compact but non-Hausdorff.
Proof. Since there are only finitely many open sets in $(X,\tau)$, this is a compact topological space. Every compact space is locally compact, since we can take $K=X$ in the definition of local compactness. The space is not Hausdorff since it contains two distinct elements $x\not=y$ (by assumption) and the only choices for $U,V$ containing $x,y$ is $U=V=X$ (since that is the only open set which is non-empty). But then $U\cap V\not=\varnothing$, so the space is not Hausdorff.
A: The discussion on local compactness on its Wikipedia page is quite relevant to this question. It discusses the three common non-equivalent (in general spaces) definitional variants of local compactness, and notes that these are all equivalent for Hausdorff ($T_2$) spaces, and this is part of the reason that Hausdorffness and local compactness are often assumed together (the property behaves better wrt operations and the combination implies Tychonoff (completely regular), and the Alexandroff compactification (one-point compactification) is better behaved, among others..)
The definition of the OP (in the link) is definition variant $2''$: every point has a local base of relatively compact neighbourhoods. 
The Alexandroff compactification of $\Bbb Q$ is an example of a $T_1$ space that is locally compact in senses $1$ and $2$ (so also loc. compact according to the OP, as $2''$ is equivalent to $2$) but not Hausdorff. That example answers the title's question.
