a topological space X
is said to be regular iff
one-point sets are closed
and if for every point x
and every closed set C
not containing x
there exist disjoint open sets U
such that x
is an element of U
is a subset of V
A space X is completely regular iff one-point sets are closed in X and if for every point x of X and every closed set C of X not containing x there exists a continuous function ƒ:X → [0, 1] such that ƒ(x) = 0 and
ƒ(c) = 1 for every c in C.
An equivalent definition of regularity is this:
A topological space X is regular iff given a point x of X and a neighborhood U of x, there is a neighborhood V of x such that the closure of V is a subset of U.
Subspaces of (completely) regular spaces are (completely) regular and a product of (completely) regular spaces is (completely) regular.
Every regular space that is second countable is normal and, furthermore, by the Urysohn metrization theorem, every regular space that is second countable is metrizable.