I realy need some help in the following questions:

1. Let $\displaystyle X$ be a topological space and let A be subgroup of X that contains one and only point from each connected component of X. Prove that $\displaystyle X/A$ is connected.

2. Let $\displaystyle X=R-{0} $ be a subspace of R. Prove that for each n=1,2,3,4... - $\displaystyle X^n$ is locally connected but $\displaystyle X^N$ is not locally connected...

Thanks a lot !