Cone space: Difference between revisions

Latest revision as of 03:00, 25 December 2010

This article describes a construct that involves some variant of taking a product of a topological space with the unit interval and then making some identifications, typically at the endpoints, based on some specific maps.
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Definition

Given a topological space $X$ , the cone space of $X$ , denoted as $CX$ , is defined as the quotient of $X\times I$ (where $I$ is the closed unit interval $[0,1]$ and we use the product topology) by the equivalence relation:

$(x_{1},0)\sim (x_{2},0)\forall x_{1},x_{2}\in X$

Here, $I$ refers to the closed unit interval $[0,1]$ .

Refer:

Cone space functor to see the properties of the cone space functor
Cone-realizable space to see the property of a topological space being realizable as the cone space over some space

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+{{interval-cum-mapping construct}}
 ==Definition==
-Given a [[topological space]] <math>X</math>, the '''cone space''' of <math>X</math>, denoted as <math>CX</math>, is defined as the quotient of <math>X \times I</math> by the equivalence relation:
+Given a [[topological space]] <math>X</math>, the '''cone space''' of <math>X</math>, denoted as <math>CX</math>, is defined as the [[defining ingredient::quotient topology|quotient]] of <math>X \times I</math> (where <math>I</math> is the [[defining ingredient::closed unit interval]] <math>[0,1]</math> and we use the [[defining ingredient::product topology]]) by the equivalence relation:
 <math>(x_1,0) \sim (x_2,0) \forall x_1,x_2 \in X</math>
+Here, <math>I</math> refers to the [[closed unit interval]] <math>[0,1]</math>.
 Refer: