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.
![[Home]](wikipedia.png)