Tychonoff space

A topological space $X$ is said to be completely regular if whenever $C\subseteq X$ is closed and $x\in X\setminus C$ then there is a continuous function $f\colon X\to[0,1]$ with $f(x)=0$ and $f(C)\subseteq\{1\}$.

A completely regular space that is also $T_{0}$ (http://planetmath.org/T0Space) (and therefore Hausdorff (http://planetmath.org/T2Space)) is called a Tychonoff space, or a $T_{3\frac{1}{2}}$ space.

Some authors interchange the meanings of ‘completely regular’ and ‘$T_{3\frac{1}{2}}$’ 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