The Separation Axioms

The Separation Axioms

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Let $X$X be a topological space. Let one-point sets in $X$X be closed.
(a) $X$X is regular if and only if given a point $x$x of $X$X and a neighborhood $U$U of $x$x there is a neighborhood $V$V of $x$x such that $\bar{V} \subset U$Vˉ⊂U
(b) $X$X is normal if and only if given a closed set $A$A and an open set $U$U containing $A$A there is an open set $V$V containing $A$A such that $\tilde{V} \subset U$V~⊂U