Hausdorffization: Difference between revisions
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==Definition== | ==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 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. | 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''', also known as '''Hausdorffification''', '''Hausdorffication''', '''maximal Hausdorff quotient''', or '''Hausdorff quotient''', 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. | ||
==References== | ==References== | ||
* [https://www.math.leidenuniv.nl/scripties/BachVanMunster.pdf The Hausdorff Quotient] by Bart Van Munster | * [https://www.math.leidenuniv.nl/scripties/BachVanMunster.pdf The Hausdorff Quotient] by Bart Van Munster | ||
Revision as of 22:37, 15 November 2015
Definition
Let be a topological space. For , define if any open set containing intersects any open set containing . The Hausdorffization, also known as Hausdorffification, Hausdorffication, maximal Hausdorff quotient, or Hausdorff quotient, of is a quotient map with the universal property that any continuous map from to a Hausdorff space factors uniquely through the Hausdorffization.
References
- The Hausdorff Quotient by Bart Van Munster