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My text book said:

Not every metrizable space is locally compact.

And it lists a counterexample as following: The subspace $Q=\{r: r=\frac pq; p,q \in Z\}$ of $R$ with usual topology, i.e., $Q$ is the set of all rationals. It said: for any open ball of any point $r \in Q$, the closure is not compact. I can't understand this sentence. Why the closures of the open balls are not compact.

Could anybody help me to understand this sentence. Thanks ahead:)

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    @ThomasE. O, OH, yes. They are equal in the metric spaces. You are right:)2012-08-01

3 Answers 3

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Let $I$ be any nontrivial interval, and $r$ be an irrational number in $I$. We want to show that $I\cap\mathbb{Q}$ is not compact in $\mathbb{Q}$. Let $O_n=(-\infty,r-1/n)\cap\mathbb{Q}$ and $U_n=(r+1/n,\infty)\cap\mathbb{Q}$. Show that $\{O_n:n\in\mathbb{N}\}\cup\{U_n:n\in\mathbb{N}\}$ is an open cover without finite subcover.

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    Yes, that works too.2012-07-31
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Another possibility: A topological space is locally compact iff it is open in its compactifications (a compactification of a topological space $X$ is a compact topological space $Y \supset X$ such that $X$ is dense in $Y$).

But $\overline{\mathbb{R}}$ is a compactification of $\mathbb{Q}$ and $\mathbb{Q}$ is not open in $\overline{\mathbb{R}}$ (in fact, the interior of $\mathbb{Q}$ is empty).

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    Your answer is very interesting. Very appreciated! Could you let me know where is the claim from or give me a link?2012-07-31
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If you are interested, $\pi$-Base (an online version of Steen and Seebach's Counterexamples in Topology) provides some more examples of metrizable spaces that are not locally compact. You can view the search result to learn more about these spaces.

Baire Space

$C[0,1]$

Cantor's Leaky Tent

Cantor's Teepee

Discrete Rational Extension of the Reals

Duncan's Space

Evenly Spaced Integer Topology

Hilbert Space

Metrizable Tangent Disc Topology

Miller's Biconnected Set

Nested Rectangles

The Irrational Numbers

The p-adic Topology on the Integers

The Post Office Metric

The Radial Metric

The Rational Numbers

Topologist's Sine Curve

Wheel Without Its Hub

$\mathbb{Z}^\mathbb{Z}$

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    Spacebook has been supplanted by [$\pi$-Base](http://topology.jdabbs.com).2014-07-29