I need the proof of the Cartesian product of Locally connected spaces is locally connected iff each space is locally connected. I do have the left implication, as it is trivial using continuity and openness of the projection aplication. But I don't know how to do the right implication. I think that it can be done using induction for n = 2 and using the bases of the locally connected spaces. The problem statment is as follows: Proof that for locally connected spaces $X_{i}$ non-empty, $X_{1} \times X_{2}... \times X_{n}$ is locally connectes iff $X_{1}, X_{2}, .. , X_{n}$ is locally connected.
Thanks in advance.