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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?

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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 $\alpha<\omega_1$.

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

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  • $\begingroup$ Thank you so much! $\endgroup$
    – Peluso
    Oct 27 at 17:08
  • $\begingroup$ Do you have a reference for Franklin's article? $\endgroup$
    – Peluso
    Oct 27 at 17:17
  • $\begingroup$ The link points to his review in Mathematical Reviews; I don't know if he published this anywhere. $\endgroup$
    – KP Hart
    Oct 27 at 17:37
  • 2
    $\begingroup$ You need a subscription to MathSciNet. I put a copy of the review here $\endgroup$
    – KP Hart
    Oct 27 at 18:10
  • 2
    $\begingroup$ That was the issue, thank you for your patience and for the copy. $\endgroup$
    – Peluso
    Oct 27 at 18:12

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