Construction of an embedding of $\mathbb{Z} \cup \{\infty\}$ into $\mathbb{R}$.

Question

Let $X$ be the one-point compactification of the integers $\mathbb{Z}$, construct an embedding of $X$ into the reals $\mathbb{R}$.

I already appreciate your hints/answers. Thanks

Answer 1

We have to make sure that the images $f(k)$ for $k\ne\infty$ form a set with no accumulation points other than $f(\infty)$. The latter should be a limit point of the points $f(k)$ when $|k|\to\infty$. I think the following does work: $$f(k):=\left\{\eqalign{2\quad&(k=0)\cr 0\quad&(k=\infty)\cr {1\over k}\quad&({\rm otherwise})\cr}\right.$$

Answer 2

As said in comments, note that the structure of $\mathbb{Z}$ like order or ring structure is completely irrelevant from topological view. $\mathbb{Z}$ is just the countable discrete space, same as say $\mathbb{N}$ or $ω$. Its one-point compactification is just $(ω + 1)$, i.e. the convergent sequence (as a space). And it is easy to find a convergent sequence in $\mathbb{R}$.