Are the Cantor set and $B=\{\frac{1}{n}|n\in Z^+\}$ homeomorphic?

Question

Is the Cantor set homeomorphic to $B=\{\frac{1}{n}|n\in Z^+\}$? So far, I have concluded both sets are bounded and closed, therefore both are compact by the Heine-Borel Theorem. Neither are path connected or connected. Also, neither have the fixed-point property. So, I posit that they are homeomorphic. Now, assuming I am correct, I am supposed to construct an explicit homeomorphism between the two sets. How do I go about starting this part? And in general, how do I go about constructing explicit homeomorphisms from the Cantor set to another set? I have trouble grasping intuitively the Cantor set itself. Thanks in advance for the help!

Answer 1

$C$, the Cantor set, is uncountable, compact and has no isolated points (no $x \in C$ such that $\{x\}$ is open in $C$).

$B$ is countable, not closed (in $\mathbb{R}$) so not compact and all of its points are isolated points.

They hardly can be more different IMHO.