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Tychonoff space
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(Definition)
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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 $T_0$ (and therefore Hausdorff) 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.
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"Tychonoff space" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: normal, T3 space
| Other names: |
Tikhonov space, Tychonoff topological space, Tikhonov topological space, Tychonov space, Tychonov topological space |
| Also defines: |
Tychonoff, completely regular, completely regular space, Tikhonov, Tychonov |
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Cross-references: Hausdorff compactification, continuous function, closed, topological space
There are 13 references to this entry.
This is version 7 of Tychonoff space, born on 2002-01-22, modified 2006-10-13.
Object id is 1534, canonical name is Tychonoff.
Accessed 10234 times total.
Classification:
| AMS MSC: | 54D15 (General topology :: Fairly general properties :: Higher separation axioms ) |
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Pending Errata and Addenda
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