Hausdorffization: Difference between revisions

Revision as of 22:35, 15 November 2015

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 a quotient map $X\to H(X)$ with the universal property that any continuous map from $X$ to a Hausdorff space factors uniquely through the Hausdorffization.

References

The Hausdorff Quotient by Bart Van Munster

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 ==Definition==
-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]]
+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 a [[quotient map]] <math>X \to H(X)</math> with the universal property that any continuous map from <math>X</math> to a Hausdorff space factors uniquely through the Hausdorffization.
-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>.
+==References==
+* [https://www.math.leidenuniv.nl/scripties/BachVanMunster.pdf The Hausdorff Quotient] by Bart Van Munster