Pathconnectedness is productclosed
From Topospaces
This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
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Contents
Statement
Propertytheoretic statement
The property of topological spaces of being pathconnected is a productclosed property of topological spaces.
Verbal statement
A product (finite or infinite) of pathconnected spaces is pathconnected 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. The continuity of this follows from the universal property of the product topology