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I have one little problem with one point compactification. Here is example: Let $Z = \alpha \mathbb N = \mathbb N \cup \{\alpha\}$ one point compactification discrete spaces $\mathbb N$ of natural number.

I know that open sets are shapes $\{\alpha\} \cup \mathbb N \backslash K$, where $K$ is compact subset of $\mathbb N$. $\mathbb N$ is discrete space so only finite subsets of $\mathbb N$ are compact.

Question: How to prove that $Z$ is compact?

Stepan Milosevic
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    Welcome. “How to show the one-point (Alexandroff) compactification is compact?” Has been asked before (by me, for instance!) and can be found in probably many different places with a Google search. I’ll bring up the post for you – FShrike Aug 29 '22 at 22:44
  • Does this answer your question? – FShrike Aug 29 '22 at 22:47

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You need to show that every open cover of the one-point compactification of $\mathbb{N}$ has a finite subcover. The open subsets of the one-point compactification are typically taken to be any open set of the original space (in this case $\mathbb{N}$), or the compactification minus a compact set of the original space (in this case $(\{\alpha\} \cup \mathbb{N}) \setminus C$, where $C$ is a finite set of $\mathbb{N}$).

Let $\mathcal{U}$ be an open cover of $\{\alpha\} \cup \mathbb{N}$. Then there must be some $U_1 \in \mathcal{U}$ such that $\alpha \in U_1$, and looking at the open sets described above, we see that this $U_1$ must exclude only finitely many points of $\{\alpha\} \cup \mathbb{N}$. Can you finish from there?

stillconfused
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  • So $U_{1}$ can be $({\alpha} \cup \mathbb N) \backslash C_{1}$, where $C_{1}$ is compact subset of $\mathb N$. I am not sure I can finish. I will try. – Stepan Milosevic Aug 29 '22 at 22:53