# Example of a weak Hausdorff space that is not Hausdorff?

I've looked on the web and haven't found a simple example.

• From Wikipedia: en.wikipedia.org/wiki/Weak_Hausdorff_space A space $X$ is weak Hausdorff if whenever $Y$ is compact Hausdorff and $f:Y\rightarrow X$ is continuous, then $f(Y)$ is closed in $X$. – Matthew Daws Feb 14 '12 at 11:16
• Welcome to MathOverflow, Prof. Solovay. Gerhard "And Happy Valentines Day, Too" Paseman, 2012.02.14 – Gerhard Paseman Feb 14 '12 at 20:15

The one-point compactification of $\mathbb{Q}$ has the property that every compact subset is closed. So it is certainly a weak Hausdorff space. But it isn't Hausdorff, as $\mathbb{Q}$ isn't locally compact.
• @Bob Solovay: Yemon Choi's definition of the one-point compactification is correct. @Yemon Choi: But your second comment is wrong, since $\mathbb{Q}$ has infinite compact subsets (e.g., any convergent sequence together with its limit forms a compact set). But every compact subset of $\mathbb{Q}$ has empty interior, so it's true that every neighbourhood of the point at infinity meets every non-empty open set. – Stephen S Feb 15 '12 at 9:05