The whole question is:
Show that the one-point-compactification of $\mathbb{N}$ (with the discrete topology) is homeomorph to the subspace $\{0\}\cup \{1/n| n \in\mathbb{N}\}$ (with the subspace topology) of $\mathbb{R}$
My Idea is that $f(n)=1/n$ is a homeomorphism between these two spaces.
It is bijective and continuous because $\mathbb{N}$ $ \cup $ $\{\infty \}$ has the discrete Topology, therefore the preimage of every open set of $\{0\}\cup\{1/n| n \in\mathbb{N}\}$ is open in the discrete topology.
We also know that $\{0\} \cup \{1/n| n \in\mathbb{N}\}$ inherits the Hausdorff-property from $\mathbb{R} $.
Therefore we have a continuous bijection from a compact space to a Hausdorff-space which is a homeomorphism.
Is this proof correct or do I still have to show that an open set around $\{\infty\}$ is still open under $f$?
Best Answer
In the one point compactification, neighborhoods of $\infty$ are compelmenets of finite subsets of $\mathbb N$. It is not true that one point compactification has discrete topolofy so you have to consider neighborhoods of $\infty$ separately.