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 $f:X\to [0,1]$ 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 $T_{\mathrm{3}\mathrm{⁤}\frac{\mathrm{1}}{\mathrm{2}}}$ 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