Tychonoff space: Difference between revisions

Revision as of 18:02, 27 January 2012

Definition

A topological space is termed a Tychonoff space if it satisfies the following equivalent conditions:

No.	Shorthand	A topological space is termed Tychonoff if ...	A topological space $X$ is termed Tychonoff if ...
1	T1 and completely regular	it is both a T1 space and a completely regular space	points are closed in $X$ , and given any point $x \in X$ and closed subset $A \subseteq X$ such that $x \notin A$ , there exists a continuous map $f : X \to [0, 1]$ such that $f (x) = 0$ and $f (a) = 1$ for all $a \in A$ .
2	Hausdorff and completely regular	it is both a Hausdorff space and a completely regular space	it is Hausdorff, and given any point $x \in X$ and closed subset $A \subseteq X$ such that $x \notin A$ , there exists a continuous map $f : X \to [0, 1]$ such that $f (x) = 0$ and $f (a) = 1$ for all $a \in A$ .
3	has a compactification	there is a compact Hausdorff space having a dense subspace (with the subspace topology) homeomorphic to it. (note: T1 assumption redundant in this case)	there is a compact Hausdorff space $C$ and a dense subspace $U$ of $X$ such that $X$ is homeomorphic to $U$ .
4	contained in compact Hausdorff	it is homeomorphic to a subspace (not necessarily dense) of a compact Hausdorff space (note: T1 assumption redundant in this case).	there is a compact Hausdorff space $C$ and a subspace $U$ of $X$ such that $X$ is homeomoephic to $U$ .

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T3.5

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 | 2 || Hausdorff and completely regular || it is both a [[Hausdorff space]] and a [[completely regular space]] || it is Hausdorff, and given any point <math>x \in X</math> and closed subset <math>A \subseteq X</math> such that <math>x \notin A</math>, there exists a [[continuous map]]<math>f:X \to [0,1]</math> such that <math>f(x) = 0</math> and <math>f(a) = 1</math> for all <math>a \in A</math>.
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 | 3 || has a [[defining ingredient::compactification]] || there is a [[defining ingredient::compact Hausdorff space]] having a dense subspace (with the [[subspace topology]]) [[homeomorphism|homeomorphic]] to it. (note: T1 assumption redundant in this case) || there is a [[compact Hausdorff space]] <math>C</math> and a dense subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomorphic to <math>U</math>.
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