I’m trying to understand the definition of locally Euclidean:
Definition: A topological space $X$ is locally Euclidean if there exists $n \in \mathbb{N}$ so that for all $x \in X$ there is a neighborhood $U$ of $x$ which is homeomorphic to an open subset $V \subset \mathbb{R} ^n$.
If $X$ and $Y$ are locally Euclidean, how would I show that $X \times Y$ is locally Euclidean? Also, how can we prove that the $n$-dimensional sphere $S^n$ is locally Euclidean?
I'm studying Introduction to Topology by Munkres.