completely Hausdorff

Definition 1.

[1]Let $\mathrm{(}X\mathrm{,}\tau \mathrm{)}$ be a topological space.Suppose that for any two different points $x\mathrm{,}y\mathrm{\in }X\mathrm{,}x\mathrm{\ne }y$,we can find two disjoint neighborhoods

$$U_x,V_y\in \tau ,x\in U_x,y\in Y_y$$

such that theirclosures are also disjoint:

$$\overline{U_x}\cap \overline{V_y}=\mathrm{\varnothing }.$$

Then we say that $\mathrm{(}X\mathrm{,}\tau \mathrm{)}$ is acompletely Hausdorff space or a $T_{\mathrm{2}\mathrm{⁤}\frac{\mathrm{1}}{\mathrm{2}}}$ space.

Notes

A synonym for functionally Hausdorff space isUrysohn space [1].Unfortunately, the definition of completely Hausdorff and $T_{2⁤\frac{1}{2}}$are not as standard as one would like since. For example, theterm completely Hausdorff space is also used to meana functionally Hausdorff space (e.g. [2]).Nevertheless, in the present convention, we have the implication:

$$\text{functionally Hausdorff}\Rightarrow \text{completely Hausdorff}\Rightarrow T_2=\text{Hausdorff},$$

which suggests why the $T_{2⁤\frac{1}{2}}$ name have been used todenote both completely Hausdorff spaces and functionally Hausdorff spaces.