Separation Axiom - Other Separation Axioms

Other Separation Axioms

There are some other conditions on topological spaces that are sometimes classified with the separation axioms, but these don't fit in with the usual separation axioms as completely. Other than their definitions, they aren't discussed here; see their individual articles.

• X is semiregular if the regular open sets form a base for the open sets of X. Any regular space must also be semiregular.
• X is quasi-regular if for any nonempty open set G, there is a nonempty open set H such that the closure of H is contained in G.
• X is fully normal if every open cover has an open star refinement. X is fully T₄, or fully normal Hausdorff, if it is both T₁ and fully normal. Every fully normal space is normal and every fully T₄ space is T₄. Moreover, one can show that every fully T₄ space is paracompact. In fact, fully normal spaces actually have more to do with paracompactness than with the usual separation axioms.
• X is sober if, for every closed set C that is not the (possibly nondisjoint) union of two smaller closed sets, there is a unique point p such that the closure of {p} equals C. More briefly, every irreducible closed set has a unique generic point. Any Hausdorff space must be sober, and any sober space must be T₀.