I was wondering whether there exists a compact Hausdorff topology on $\mathbb N$. The only result I was able to find in this context was that, if a set has a topology that is compact, Hausdorff and has no isolated points then the set is uncountable. But what if isolated points are allowed?
Does there exist a compact Hausdorff topology on the natural numbers?
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7Identify $\mathbb{N}$ with a convergent sequence and its limit point, say in $\mathbb{R}$. – 2011-08-31
2 Answers
Of course there is.
Example: Consider $f\colon\mathbb N\to\mathbb R$ defined as, $f(0)=0, f(n)=\frac1n$. Now let $\tau$ be the topology defined as:
$U\subseteq\mathbb N$ is open if and only if $f''U$ is open in $\mathbb R$.
This clearly corresponds to the metric $d(x,y)=|f(x)-f(y)|$, so this makes $(\mathbb N,\tau)$ a metrizable space. In fact it is completely metrizable since every point except $0$ is isolated, and $0$ is the only limit point.
This can be easily generalized by taking a countable set of real numbers which is bounded and has only countably many limit points, and any bijection whatsoever between this set and $\mathbb N$.
Of course we cannot drop the limitation that there are only countably many limit points, since a compact metric space is complete and all limit points must be inside it.
If you are familiar with ordinals, then you may want to prove the following:
Theorem: Suppose $\beta$ is a successor ordinal, then in the order topology $\beta$ is Hausdorff and compact.
Corollary: Every countable successor ordinal can be given a compatible metric which is complete, and the result is a compact Polish space (separable, metric and complete).
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0By $f''(U)$, do you mean $f(U)$? – 2011-08-31
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0@Zev: Yes. $f''U=\{f(x)\mid x\in U\}$. It is a common notation (in set theory). – 2011-08-31
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0Interesting; I will remember it. Thanks! – 2011-08-31
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0@Zev: I was wondering about that, too. Asaf: Why two primes? What is $f'U$, then? The pre-image of $U$? – 2011-08-31
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0@Theo: I don't recall seeing $f'U$ anywhere in set theory, so I can't really tell you that. Thing is that when the function has an unspecified domain, writing $f(U)$ could be the image of the element $U$ (if the function is defined on both $U$ as an element and as a set), so there are several notations $f''U, f[U], \operatorname{Rng}(f|_U)$, etc). – 2011-08-31
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0Thanks. It seems to me that $\operatorname{Rng}(f|_U)$ is about as complicated as what it is supposed to abbreviate... – 2011-08-31
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0@Theo: What do you think it is supposed to abbreviate? I always though that $f''U$ abbreviates $\operatorname{Rng}(f|_U)$ to begin with :-) – 2011-08-31
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0Not the other way around? :) I count 10 symbols in $\{f(x):x \in U\}$ while $\operatorname{Rng}(f|_U)$ contains 8. Not to mention the texed version... – 2011-08-31
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0@Theo: Oh, yeah. If you think about it this way. I always thought about the former as the definition of the latter, and $f''U$ as the abbreviation thereof :-) – 2011-08-31
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0You know, I'm still too much of an analyst that I could ever consider abusing primes this way... Anyway, I guess *de gustibus...* – 2011-08-31
There exists a bijection between the one-point compactification $\mathbb{N}\cup\lbrace\infty\rbrace$ and $\mathbb N$, for instance by mapping
$$ \infty \mapsto 0 \text{ and } n \mapsto n+1 .$$
Use this map to obtain a compact Hausdorff topology on $\mathbb N$.