one point compactification of discrete space

Question

Problem: What is the one point compactification $X^*$ of a discrete space $X$.

In the case of $X$ being finite, $X$ itself is compact so the one point compactification would be merely $X$ $\bigcup$ {$\infty$}.

Now in the case of $X$ being infinite I need to consider two cases.

When $X$ is countably infinite, I have shown that $X^*$ is homeomorphic to {$0$} $\bigcup$ {$1/n$ | $n \in N$} in the subspace topology of the usual $R$.

However, in the case of $X$ being uncountable, I cannot come up with any familiar space that $X^*$ is homeomorphic to. Can anyone help me out?

Answer 1

$X^*=X\cup \{\infty\}$ where the open neighbourhoods of $\infty$ are the cofinite sets.

Answer 2

One point compactification of discrete topological space is a profinite space. By definition, profinite space is inverse limit of finite discrete topological space. This is equivalent to say compact Hausdorff and totally disconnected. I recommend the book "profinite groups" by Ribes and Zalesskii, page 89.