Path-connectedness is product-closed

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This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
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Statement

Property-theoretic statement

The property of topological spaces of being 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. The continuity of this follows from the universal property of the product topology