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Munkres says of the necessity of the Hausdorff Condition in the Proposition "Every compact supsapce of a Hausdorff space is closed":

One needs the Hausdorff condition in the hypothesis. Consider, the finite complement topology on $\mathbb{R}$. The only proper subsets of $\mathbb{R}$ that are closed in this topology are finite sets. But every subset of $\mathbb{R}$ is compact in this topology.

I'm having trouble seeing this. In particular, how does one build sets in this topology that are anything but $\mathbb{R}-\{x_0,...x_n\}$? In order to get a finite set, I need to intersect uncountably many sets like this -- which I can't do in a topology. I can see that every subset of $\mathbb{R}$ will be compact -- but not every subset is attainable in this topology. So this statement confuses me.

I guess what I'm looking for are examples of both statements.

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    $\begingroup$ What do you mean "not every subset is attainable"? In general, you can have sets that are open, closed, both or neither. The statement does not claim that all sets are open for this topology. $\endgroup$ – Randall Sep 27 '18 at 14:23
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    $\begingroup$ And, $[0,1]$ is still a set that I can "build" in $\mathbb{R}$ with this topology, it's just not open and not closed, but it is compact. $\endgroup$ – Randall Sep 27 '18 at 14:24
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There is some confusion in what you wrote. If I have a topology $\tau$ on $\mathbb R$ and I want to defined a certain subset of $\mathbb R$, then $\tau$ doesn't matter. It is only a matter of defining the set. And, for instance, $[-1,1]$ is compact subset of $\mathbb R$ with respect to finite complement topology, in spite of the fact that it is not closed.

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  • $\begingroup$ ah okay! So I can define any subset - but assessing whether such a subset is open or closed will have to do with the topology. I can see compactness of $[-1,1]$. How do I assess closedness? $\endgroup$ – yoshi Sep 27 '18 at 14:36
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    $\begingroup$ @yoshi It is not closed because, for this topology, the only infinite closed set is $\mathbb R$ itself. $\endgroup$ – José Carlos Santos Sep 27 '18 at 14:37

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