Contractibility is product-closed - Revision history

Vipul at 11:21, 8 August 2008

2008-08-08T11:21:48Z

← Older revision		Revision as of 11:21, 8 August 2008
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	{{topospace metaproperty satisfaction}}		{{topospace metaproperty satisfaction\|
			property = contractible space\|
			metaproperty = product-closed property of topological spaces}}

	==Statement==		==Statement==

Vipul: /* Generic proof (for an arbitrary family) */

2008-07-20T19:33:43Z

Generic proof (for an arbitrary family)

← Older revision		Revision as of 19:33, 20 July 2008
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	'''Given''': An indexing set <math>I</math>, a collection <math>\{ X_i \}_{i \in I}</math> of [[contractible space]]s. <math>X</math> is the product of the <math>X_i</math>s, endowed with the [[product topology]]		'''Given''': An indexing set <math>I</math>, a collection <math>\{ X_i \}_{i \in I}</math> of [[contractible space]]s. <math>X</math> is the product of the <math>X_i</math>s, endowed with the [[product topology]]

	'''To prove''': <math>P</math> is a contractible space		'''To prove''': <math>X</math> is a contractible space

	'''Proof''': Since each <math>X_i</math> is contractible, we can choose, for each <math>X_i</math>, a point <math>p_i \in X_i</math>, and a contracting homotopy <math>F_i: X_i \times [0,1] \to X_i</math>, with the property that:		'''Proof''': Since each <math>X_i</math> is contractible, we can choose, for each <math>X_i</math>, a point <math>p_i \in X_i</math>, and a contracting homotopy <math>F_i: X_i \times [0,1] \to X_i</math>, with the property that:

Vipul: /* Proof */

2008-07-20T19:33:19Z

Proof

← Older revision		Revision as of 19:33, 20 July 2008
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	===Generic proof (for an arbitrary family)===		===Generic proof (for an arbitrary family)===

	'''Given''': An indexing set <math>I</math>, a collection <math>\{ X_i \}_{i \in I}</math> of [[contractible space]]s. <math>P</math> is the product of the <math>X_i</math>s, endowed with the [[product topology]]		'''Given''': An indexing set <math>I</math>, a collection <math>\{ X_i \}_{i \in I}</math> of [[contractible space]]s. <math>X</math> is the product of the <math>X_i</math>s, endowed with the [[product topology]]

	'''To prove''': <math>P</math> is a contractible space		'''To prove''': <math>P</math> is a contractible space

Vipul at 19:24, 20 July 2008

2008-07-20T19:24:19Z

← Older revision		Revision as of 19:24, 20 July 2008
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	==Statement==		==Statement==

	===~~For two spaces~~===		===Property-theoretic statement===

	~~Let <math>X</math> and <math>Y</math> be~~ [[contractible space]]~~s. Then~~ the product ~~space <math>X \times Y</math> is contractible~~.		The [[property of topological spaces]] of being a [[contractible space]], satisfies the [[metaproperty of topological spaces]] of being [[product-closed property of topological spaces\|product-closed]].

			===Statement with symbols===

	~~===For an arbitrary family of spaces===~~
	Let <math>X_i</math>, <math>i \in I</math>, be an indexed family of topological spaces. Then the product space, endowed with the [[product topology]], is contractible.		Let <math>X_i</math>, <math>i \in I</math>, be an indexed family of topological spaces. Then the product space, endowed with the [[product topology]], is contractible.

Vipul at 19:22, 20 July 2008

2008-07-20T19:22:33Z

@@ Line 3: / Line 3: @@
 ==Statement==
-===Symbolic statement===
+===For two spaces===
 Let <math>X</math> and <math>Y</math> be [[contractible space]]s. Then the product space <math>X \times Y</math> is contractible.
 ==Proof==
-===Key idea===
+===Key idea (for two spaces)===
 Suppose <math>F: X \times I \to X</math> and <math>G:Y \times I \to Y</math> are contracting homotopies for <math>X</math> and <math>Y</math>. Then the map <math>F \times G</math> defined as:
@@ Line 20: / Line 21: @@
 Thus <math>X \times Y</math> is contractible.

Vipul: 4 revisions

2008-05-11T19:42:32Z

4 revisions

← Older revision	Revision as of 19:42, 11 May 2008
(No difference)

Vipul at 01:13, 27 December 2007

2007-12-27T01:13:23Z

← Older revision		Revision as of 01:13, 27 December 2007
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			{{topospace metaproperty satisfaction}}

	==Statement==		==Statement==

			===Symbolic statement===
	Let <math>X_i</math>, <math>i \in I</math>, be an indexed family of topological spaces. Then the product space, endowed with the [[product topology]], is contractible.		Let <math>X_i</math>, <math>i \in I</math>, be an indexed family of topological spaces. Then the product space, endowed with the [[product topology]], is contractible.

Vipul at 21:04, 15 December 2007

2007-12-15T21:04:01Z

← Older revision		Revision as of 21:04, 15 December 2007
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	==Statement==		==Statement==

			Let <math>X_i</math>, <math>i \in I</math>, be an indexed family of topological spaces. Then the product space, endowed with the [[product topology]], is contractible.

			We describe the proof for two spaces; the same idea works in general:
	Let <math>X</math> and <math>Y</math> be [[contractible space]]s. Then the product space <math>X \times Y</math> is contractible.		Let <math>X</math> and <math>Y</math> be [[contractible space]]s. Then the product space <math>X \times Y</math> is contractible.

Vipul: Contractibility is direct product-closed moved to Contractibility is product-closed

2007-12-15T21:02:26Z

Contractibility is direct product-closed moved to Contractibility is product-closed

← Older revision	Revision as of 21:02, 15 December 2007
(No difference)

Vipul at 22:17, 26 September 2007

2007-09-26T22:17:20Z

New page

==Statement==

Let <math>X</math> and <math>Y</math> be [[contractible space]]s. Then the product space <math>X \times Y</math> is contractible.

==Proof==

===Key idea===

Suppose <math>F: X \times I \to X</math> and <math>G:Y \times I \to Y</math> are contracting homotopies for <math>X</math> and <math>Y</math>. Then the map <math>F \times G</math> defined as:

<math>(F \times G)(x,y,t) = (F(x,t),G(y,t))</math>

is a contracting homotopy for <math>X \times Y</math>.

Thus <math>X \times Y</math> is contractible.