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I want to know in a Hausdorff topological space, whether separation axiom holds for disjoint closed sets $A_1,A_2$? i.e. given $A_1\cap A_2=\emptyset$ both closed set, is the following statement true $\exists G_1,G_2, s.t. \ A_1\subset G_1,A_2\subset G_2$ where $G_1$ and $G_2$ are open sets and $G_1 \cap G_2 =\emptyset$

What if the space is not only Hausdorff topological but also compact?

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1 Answer 1

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(I assume you meant $G_1$ and $G_2$ to be disjoint.)

No. A Hausdorff space satisfying this axiom is called normal or $T_4$ (depending on the author, a non-Hausdorff space with this property might also be called normal, but not $T_4$). A Hausdorff space need not be normal in general.

A compact Hausdorff space is always normal. The canonical example of a non-normal Hausdorff (even completely regular Hausdorff) space is the Niemytzki plane.

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