products of connected spaces are connected

Theorem 1

[1, 2] Let ${\mathrm{(}{X}_i\mathrm{)}}_{i\mathrm{\in }I}$ be a family of topological spaces. Then the product space

$$\prod _{i\in I}{X}_i$$

with the product topology is connected if and only if each space $X_i$ is connected.

As is true of most results in topology involving products, the forward implication requires the axiom of choice.

Title	products of connected spaces are connected
Canonical name	ProductsOfConnectedSpacesAreConnected
Date of creation	2013-03-22 13:56:13
Last modified on	2013-03-22 13:56:13
Owner	mps (409)
Last modified by	mps (409)
Numerical id	6
Author	mps (409)
Entry type	Theorem
Classification	msc 54D05