Wedge sum - Revision history

Vipul at 16:50, 30 December 2010

2010-12-30T16:50:27Z

← Older revision		Revision as of 16:50, 30 December 2010
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	The wedge sum is the (unique in the usual sense) [[coproduct]] in the [[category of based topological spaces]]. In other words, given a bunch of topological spaces <math>(X_i,x_i), i \in I</math>, and [[continuous map]]s <math>f_i:(X_i,x_i) \to (Z,z)</math> for some [[based topological space]] <math>(Z,z)</math>, there is a unique continuous map <math>f</math> from the wedge sum to <math>(Z,z)</math> such that composing <math>f</math> with the respective inclusion maps gives <math>f_i</math>.		The wedge sum is the (unique in the usual sense) [[coproduct]] in the [[category of based topological spaces]]. In other words, given a bunch of topological spaces <math>(X_i,x_i), i \in I</math>, and [[continuous map]]s <math>f_i:(X_i,x_i) \to (Z,z)</math> for some [[based topological space]] <math>(Z,z)</math>, there is a unique continuous map <math>f</math> from the wedge sum to <math>(Z,z)</math> such that composing <math>f</math> with the respective inclusion maps gives <math>f_i</math>.

			==Facts==

			===Wedge sum is associative===

			Since the wedge sum is a coproduct, it is associative in the category of based topological spaces.

			However, when the wedge sum is viewed over topological spaces ''without'' specification of basepoint, it is not uniquely defined, and hence different ways of associating it may correspond to different interpretations.

			For instance, the [[wedge of three circles]] has two different interpretations, depending on whether we think of wedging all three circles together, or whether we wedge two circles at one basepoint and then choose a different basepoint in the [[wedge of two circles]] to wedge with the third circle.

Vipul at 16:45, 30 December 2010

2010-12-30T16:45:14Z

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 ==Definition==
-Given two topological spaces with basepoints: <math>(X,x_0)</math> and <math>(Y,y_0)</math>, their wedge sum is defined as the quotient of their disjoint union by the identification <math>x_0 \sim y_0</math>.
+===For two spaces===
-In other words, it is the union of the spaces, identified at the points <math>x_0</math> and <math>y_0</math>.
+Given two [[defining ingredient::based topological space]]s: <math>(X,x_0)</math> and <math>(Y,y_0)</math>,

Vipul: 1 revision

2008-05-11T20:00:06Z

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Vipul at 14:02, 22 May 2007

2007-05-22T14:02:36Z

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{{product notion for topospaces}}

==Definition==

Given two topological spaces with basepoints: <math>(X,x_0)</math> and <math>(Y,y_0)</math>, their wedge sum is defined as the quotient of their disjoint union by the identification <math>x_0 \sim y_0</math>.

In other words, it is the union of the spaces, identified at the points <math>x_0</math> and <math>y_0</math>.