Example of compact non-Hausdorff space Can you give an example of a compact non-Hausdorff space with a point which has no local base formed by compact neighborhoods?
Thank you in advance.
 A: The one-point compactification of the rationals is such a space: $X = \mathbb{Q} \cup \{\infty\}$ , where neighbourhoods of rational $x$ are the usual ones (inherited from $\mathbb{R}$) and neighbourhoods of $\infty$ are of the form $\{\infty\} \cup \mathbb{Q}\setminus K$, where $K$ is a compact subset of $\mathbb{Q}$ (in the usual topology). 
All rational points still don't have a base of compact neighbourhoods (every open interval intersected with the rationals contain sequences of rationals converging to an irrational (in the  reals) so that sequential compactness fails), while $X$ is compact by construction, but not Hausdorff (as the one-point compactification of a space $Y$ is Hausdorff iff $Y$ is locally compact Hausdorff). 
So this is actually a family of examples: one-point compactifications of non-locally compact spaces $(X, \mathcal{T}_X)$, defined in the same way in general: the set $\alpha X = X \cup \{\infty\}$ with topology 
$$\mathcal{T}_X \cup \{ (X\setminus K) \cup \{\infty\}: \text{ with } K \subseteq X \text{ closed and compact }\}$$.
$\alpha X$ is always compact (same proof as for locally compact spaces), $X$ is dense in $\alpha X$ and $x \rightarrow x$ is an open embedding  of $X$ into $\alpha X$. $\alpha X$ is Hausdorff iff $X$ is locally compact and Hausdorff.
(see wikipedia, e.g.). It's because of the last property that it's usually only introduced (in books) for the locally compact case. (as compact Hausdorff spaces have nicer properties than compact and $T_1$ spaces).
