Proof That the Cartesian Product of a Family of Locally Connected Spaces is Locally Connected
Theorem
If
is a family of locally connected spaces, then
is locally connected.
Proof
Let
and let
be a neighbourhood of
A member
of the base forexists such that
and
Each coordinate space is locally connected hence connected open subsets
exist such that
and
for
Let
is connected because each
andis connected. Sinceis open and
is locally connected.
