Path-connectedness is product-closed - Revision history

Vipul: 2 revisions

2008-05-11T19:57:28Z

2 revisions

← Older revision	Revision as of 19:57, 11 May 2008
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Vipul: /* Proof outline */

2007-12-27T01:09:42Z

Proof outline

← Older revision		Revision as of 01:09, 27 December 2007
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	* Start with the two points; write them down as tuples		* Start with the two points; write them down as tuples
	* In each coordinate, construct a path from that coordinate for one point, to that coordinate for the other point		* In each coordinate, construct a path from that coordinate for one point, to that coordinate for the other point
	* Take the path, which in a given coordinate, is the path constructed above for that coordinate		* Take the path, which in a given coordinate, is the path constructed above for that coordinate. The continuity of this follows from the universal property of the product topology

Vipul at 01:09, 27 December 2007

2007-12-27T01:09:13Z

New page

{{topospace metaproperty satisfaction}}

==Statement==

===Property-theoretic statement===

The [[property of topological spaces]] of being [[path-connected space|path-connected]] is a [[product-closed property of topological spaces]].

===Verbal statement===

A product (finite or infinite) of path-connected spaces is path-connected in the [[product topology]].

==Proof==

===Proof outline===

We need to show that any two points in the product can be joined by a path:

* Start with the two points; write them down as tuples
* In each coordinate, construct a path from that coordinate for one point, to that coordinate for the other point
* Take the path, which in a given coordinate, is the path constructed above for that coordinate