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Tychonoff space (Definition)

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.



"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
Keywords:  topology
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Cross-references: Hausdorff compactification, continuous function, closed, topological space
There are 11 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 6682 times total.

Classification:
AMS MSC: 54D15 (General topology :: Fairly general properties :: Higher separation axioms )
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Tychonov by Wkbj79 on 2006-10-12 01:00:24
The spelling that I have seen most frequently is "Tychonov".

I decided to post about this rather than file a correction for many reasons. I searched, and there are 4 entries that use "Tychonoff". If I were to file a correction to this entry, I would feel compelled to file similar corrections to the other entries if applicable, and I didn't want to search these and all other alternative spellings, check these entries to see whether the spelling "Tychonov" is included or not, then file an addendum to each of these entries suggesting that the author add "Tychonov". I thought that it would be a much better idea to post one message so that everyone who owns an entry with one of the many other spellings occurring in the entry and decide for himself or herself whether to add "Tychonov" to the synonyms section.

I hope that this makes sense to anyone besides me. :-)

Warren

PS--The spellchecker thinks both "Tychonoff" and "Tychonov" are misspellings. LOL!
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