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Recently, I'm reading Engelking's book. In this book, it always uses as the example the space $A(m)$, see for instance the example 1.1.8, page 15.


For this space, I'm not very clear, for the description is complex to me. Sometimes I view it as a sequence with a limit point when $m=\omega$. But I am not sure I'm completely right. So, could anybody knowing this example help me to describe it more clearly? Thanks ahead.

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It would help if you could scan or type the description, in case anyone doesn't own the book. –  Kevin Carlson Aug 13 '12 at 3:07
@Paul It denotes the one-point compactification of the discrete space of cardinality $m$. –  azarel Aug 13 '12 at 3:09
@KevinCarlson I try to find the link for the book, however, I cannot find it now. –  Paul Aug 13 '12 at 3:15
@azarel Why does it denote the one-point compactification? Could you explain more? It will be welcome if you write it as an answer:) –  Paul Aug 13 '12 at 3:17
Compare this topology with the construction given in the Alexandroff Compactification Theorem (3.5.11, pp.169-170) on an infinite discrete space. (If $X_0$ denotes the subspace $X \setminus \{ x_0 \}$, note that $X_0$ is discrete, and then look at what the neighbourhoods of $x_0$ are.) –  Arthur Fischer Aug 13 '12 at 3:28

1 Answer 1

up vote 1 down vote accepted

Let $D$ be the discrete space of cardinality $\kappa$, and let $x_0$ be a point not in $D$. Let $A=D\cup\{x_0\}$ be the one-point compactification of $D$, with $x_0$ as the added point. The compact subsets of $D$ are precisely the finite sets, so open nbhds of $x_0$ in $A$ have the form $\{x_0\}\cup(D\setminus F)$ for finite $F\subseteq D$. Let $\mathcal{O}$ be the topology on $A$; what, exactly, is $\mathcal{O}$?

Let $S\subseteq A$. If $x_0\notin S$, then $S\subseteq D$, so $S\in\mathcal O$ iff $S$ is open in $D$. But $D$ is discrete, so $S$ is open in $D$ $-$ all subsets of $D$ are open in $D$ $-$ and therefore $S\in\mathcal{O}$. If $x_0\in S$, then $S$ must contain an open nbhd of $x_0$, so there must be a finite $F\subseteq D$ such that $\{x_0\}\cup(D\setminus F)\subseteq S$. But then $$A\setminus S\subseteq A\setminus\Big(\{x_0\}\cup(D\setminus F)\Big)=D\setminus(D\setminus F)=F\;,$$ so $A\setminus S$ is finite. In other words, $S\in\mathcal O$ iff either $x_0\notin S$, or $x_0\in S$ and $A\setminus S$ is finite. Finally, this is the same as saying that $S\in\mathcal O$ iff $x_0\notin S$, or $A\setminus S$ is finite, which is exactly the definition of the topology $\mathcal O$ given in Example 1.1.8. Thus, $A(\kappa)$ is, as azarel said, the one-point compactification of the discrete space of cardinality $\kappa$.

$A(\omega)$ is indeed homeomorphic to $\omega+1$, a simple sequence with a limit point. If $\kappa>\omega$, it’s similar, but it’s not a sequence: it’s a set of $\kappa$ isolated points and one limit point, $x_0$, with the property that every open nbhd of $x_0$ contains all but finitely many of the isolated points. This means, among other things, that if $S$ is an infinite set of isolated points, then $x_0\in\operatorname{cl}S$: every open nbhd of $x_0$ must contain all but finitely many points of $S$. It also means that $A(\kappa)$ is compact: once you’ve covered the point $x_0$, there are at most finitely many points left to be covered. (Of course this is also clear from the fact that $A(\kappa)$ is a one-point compactification!)

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