Tychonoff space

A topological spaceMathworldPlanetmath X is said to be completely regularPlanetmathPlanetmath if whenever CX is closed and xXC then there is a continuous functionMathworldPlanetmathPlanetmath f:X[0,1] with f(x)=0 and f(C){1}.

A completely regular space that is also T0 (http://planetmath.org/T0Space) (and therefore HausdorffPlanetmathPlanetmath (http://planetmath.org/T2Space)) is called a Tychonoff space, or a T312 space.

Some authors interchange the meanings of ‘completely regular’ and ‘T312’ compared to the above.

It can be proved that a topological space is Tychonoff if and only if it has a Hausdorff compactification.

Title Tychonoff space
Canonical name TychonoffSpace
Date of creation 2013-03-22 12:12:42
Last modified on 2013-03-22 12:12:42
Owner yark (2760)
Last modified by yark (2760)
Numerical id 11
Author yark (2760)
Entry type Definition
Classification msc 54D15
Synonym Tikhonov space
Synonym Tychonoff topological space
Synonym Tikhonov topological space
Synonym Tychonov space
Synonym Tychonov topological space
Related topic NormalTopologicalSpace
Related topic T3Space
Defines Tychonoff
Defines completely regular
Defines completely regular space
Defines Tikhonov
Defines Tychonov