product of path connected spaces is path connected

PropositionPlanetmathPlanetmathPlanetmath. Let X and Y be topological spacesMathworldPlanetmath. Then X×Y is path connected if and only if both X and Y are path connected.

Proof. ”” Assume that X and Y are path connected and let (x1,y1),(x2,y2)X×Y be arbitrary points. Since X is path connected, then there exists a continous map


such that


Analogously there exists a continous map


such that


Then we have an induced map


defined by the formulaMathworldPlanetmathPlanetmath:


which is continous path from (x1,y1) to (x2,y2).

” On the other hand assume that X×Y is path connected. Let x1,x2X and y0Y. Then there exists a path


such that


We also have the projection map π:X×YX such that π(x,y)=x. Thus we have a map


defined by the formula


This is a continous path from x1 to x2, therefore X is path connected. Analogously Y is path connected.

Title product of path connected spaces is path connected
Canonical name ProductOfPathConnectedSpacesIsPathConnected
Date of creation 2013-03-22 18:31:02
Last modified on 2013-03-22 18:31:02
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Theorem
Classification msc 54D05