Hausdorffization: Difference between revisions

Revision as of 21:22, 26 December 2007

Definition

Let $X$ be a topological space. For $a,b\in X$ , define $a\sim b$ if any open set containing $a$ intersects any open set containing $b$ . The Hausdorffization of $X$ is the quotient map of $X$ by the equivalence relation generated by $\sim$ . The term Hausdorffization is also sometimes used for the quotient space obtained after taking the map. Clearly, this is a Hausdorff space

The Hausdorffization has the following universal property: any map from $X$ to a Hausdorff space factors through the Hausdorffization of $X$ .

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 ==Definition==
-Let <math>X</math> be a [[topological space]]. For <math>a,b \in X</math>, define <math>a \simeq b</math> if any open set containing <math>a</math> intersects any open set containing <math>b</math>. The '''Hausdorffization''' of <math>X</math> is the [[quotient map]] of <math>X</math> by the equivalence relation generated by <math>\simeq</math>. The term '''Hausdorffization''' is also sometimes used for the quotient space obtained after taking the map. Clearly, this is a [[Hausdorff space]]
+Let <math>X</math> be a [[topological space]]. For <math>a,b \in X</math>, define <math>a \sim b</math> if any open set containing <math>a</math> intersects any open set containing <math>b</math>. The '''Hausdorffization''' of <math>X</math> is the [[quotient map]] of <math>X</math> by the equivalence relation generated by <math>\sim</math>. The term '''Hausdorffization''' is also sometimes used for the quotient space obtained after taking the map. Clearly, this is a [[Hausdorff space]]
 The Hausdorffization has the following ''universal property'': any map from <math>X</math> to a [[Hausdorff space]] factors through the Hausdorffization of <math>X</math>.