Separation axiom: Encyclopedia II - Separation axiom - Other separation axioms

Separation axiom - 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 won't be discussed here.

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 fully normal if every open cover has an open star refinement. Every fully normal space must also be both normal regular and paracompact. In fact, fully normal spaces actually have more to do with paracompactness than with the usual separation axioms.

X is fully T₄, or fully normal Hausdorff, if it is both T₁ and fully normal. A fully T₄ space must also be T₄.

X is sober if, for every closed set C which 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.

Adapted from the Wikipedia article "Other separation axioms", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki