Can an open set in $\mathbb{R}^n$ homeomorphic to a non-open set? Suppose $A$ is an open set of $\mathbb{R}^n$ under the ordinary topology and is homeomorphic to $B\subset \mathbb{R}^n$. Is it necessary that $B$ be open too?

Yes, but this is very specific to $\Bbb R^n$ and doesn't hold in arbitrary spaces. It's a highly non-trivial fact (due to Brouwer) called the invariance of domain (see here e.g.); here domain is an old school word for open subset of $\Bbb R^n$.
You cannot really prove it easily without some machinery (Brouwer's fixed point theorem, or some dimension theory; nowadays most will approach it via algebraic topology methods).
