Tychonoff space

A Hausdorff space X is called a Tychonoff space if,

for every nonempty closed subset C and every x in the complement of C,

there is a continous function

f : X -> [0,1]

such that f(x) = 0 and f(C) = {1}.

Tychonov spaces are also called T_{3 1/2} spaces,

T_π spaces or completely regular spaces,

although these other terms are sometimes used for non-Hausdorff spaces

with the above property.

Note that Tychonoff is a Russian name and there are several ways to transliterate it. Alternatives include Tychonov, Tikhonov, Tihonov and Tichonov.

Tychonoff spaces are precisely those topological spaces which can be

embedded in a compact Hausdorff space.

Examples of Tychonoff spaces include:

• All metric spaces.

• All locally compact Hausdorff spaces, and therefore all manifolds.

• All linearly ordered topological spaces.

• All Hausdorff topological groups.

• All CW complexes.

• All products of Tychonoff spaces.

• All subspaces of Tychonoff spaces.