Product of Mobius Band with Itself

Question

I have a few questions about the Mobius band $M$. The first two questions are pretty direct, whereas the third is a bit vague.

1. Let $F_2(M)$ denote the collection of non-empty subsets of $M$ with at most two points. What is the topology of this space under the Hausdorff metric?

Generally, if $X$ is a metric space let $F_n(X)$ be the non-empty subsets of $X$ with at most $n$ points, endowed with the Hausdorff metric. For example, $F_2(\mathbb{S}^1) = M$.

2. What is $F_4(\mathbb{S}^1)$?

3. Is there a study of the space $M \times M$? What are some of its most interesting and important properties?

Thanks!