Proof That the Cartesian Product of a Family of Locally Connected Spaces is Locally Connected

Theorem

Ifis a family of locally connected spaces, thenis locally connected.

Proof

Letand letbe a neighbourhood ofA memberof the base forexists such thatandEach coordinate space is locally connected hence connected open subsetsexist such thatandfor

Let

is connected because eachandis connected. Sinceis open andis locally connected.