Are spaces $ A = \left\{ \frac{1}{n} \,\middle|\, n \in \mathbb{N} \right\} $ and $ B = \left\{ \frac{n+1}{n} \,\middle|\, n \in \mathbb{N} \right\} $ as subspaces of $ ( 0 , + \infty)$ homeomorphic? Are $\operatorname{Cl}(A)$ and $\operatorname{Cl}(B)$ homeomorphic?
I would say that $A$ and $B$ aren't homeomorphic because $A$ is compact and $B$ is not compact (continuous function invariants). But when it comes to $\operatorname{Cl}(A)$ and $\operatorname{Cl}(B)$ they are both compact so it would seem that they might be homeomorphic, but how do I construct the homeomorphism? The problem is the element $1$ of $B$, I can't "send" any element of A to it.
Yes, they are homeomorphic. Just consider$$\begin{array}{rccc}f\colon&A&\longrightarrow&B\\&x&\mapsto&x+1;\end{array}$$it's a homeomorphism.
However, $\overline A$ and $\overline B$ are not homeomorphic, since $\overline A$ isn't compact, whereas $\overline B$ is.