Proof That a Product of Connected Spaces is Connected

Theorem

A product of connected spaces is connected.

Proof

Letbe a collection of connected spaces and letbe the product space.

Letand letbe the component to whichbelongs.

Takeand letbe any open set containing

The setis homeomorphic tohence is connected.

Sincewhereis the component of

The sethence

Thenhas one component and is connected.