In [Between $T\_1$ and $T\_2$](https://doi.org/10.2307/2316017), Albert Wilansky mentioned in **6.** that it was not known whether or not every locally compact US space is $T\_2$.


Is this matter still an open problem?



Is this question about US spaces still an open problem?
No, it is not. S. Franklin gave an example in his [review of Wilansky's paper](https://mathscinet.ams.org/mathscinet-getitem?mr=208557): take a compact space with a point $x$ that is not the limit of a non-trivial sequence, for example the ordinal $\omega\_1+1$ with $x=\omega\_1$, and double that point. The set of our space is $\omega\_1\cup\{\omega\_1^a, \omega\_1^b\}$; the points other than $\omega\_1^a$ and $\omega\_1^b$ have their normal neighbourhoods and the (basic) neigbourhoods of $\omega\_1^i$ ($i=a,b$) are $(\alpha,\omega\_1)\cup\{\omega\_1^i\}$ with $\alpha<\omega\_1$.


The result is US and (locally compact), but not Hausdorff.


