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Let $X$ be a topological space. Assume that for all $x_1,x_2 \in X$ there exist open neighbourhoods $U_i$ of $x_i$ such that $U_1 \cap U_2 = \emptyset$. Such a space, as we all know, is called Hausdorff. What would we call a space, and which separation axioms would the space satisfy, if $\overline{U_1} \cap \overline{U_2} = \emptyset$ for all $x_i \in X$?

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Such a space is known as $T_{2\frac{1}{2}}$ or Urysohn according to Wikipedia.

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    $\begingroup$ Seriously? 5 seconds apart? :-) $\endgroup$ – Asaf Karagila Aug 31 '12 at 0:31
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    $\begingroup$ @Asaf You just have to rub it in my face... ;-) $\endgroup$ – Michael Greinecker Aug 31 '12 at 0:32
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    $\begingroup$ I'll let you have this one... $\endgroup$ – Asaf Karagila Aug 31 '12 at 0:32

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