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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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up vote 2 down vote accepted

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

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

These spaces fall on a continuum, Hausdorff (T2), Urysohn (T2 1/2), completely Hausdorff (T3), and Tichonoff (T3 1/2)

But the one that you are referring to is a Urysohn (T2 1/2) space. As such, it is closed neighborhoods that separate any two distinct points.

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