# Is this question about US spaces still an open problem?

In Between $$T_1$$ and $$T_2$$, 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?

No, it is not. S. Franklin gave an example in his review of Wilansky's paper: 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 $$\alpha<\omega_1$$.