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 Tychonoffif and only if it has a Hausdorff compactification.

单词	TychonoffSpace
释义	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 Tychonoffif and only if it has a Hausdorff compactification.
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