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Is this Locally Compact?

  • Thread starter talolard
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  • #1
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Homework Statement




Is the space [tex] X=\mathbb{R\backslash}\left\{ a+b\sqrt{2}\::\: a,b\in\mathbb{Q}\right\} [/tex] locally compact?

Homework Equations


According to the baire theorem, in a locally compact hausdorf space, the intersection of dense open sets is dense.

The Attempt at a Solution


I'm leaning for a no here and want to show a violation of the baire theorem. I'm not sure how to go about constucting my dense sets because I am not sure what is dense in this subset.
Thanks
Tal
 

Answers and Replies

  • #2
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Hmm, nice try. But I think that the intersection of dense subsets IS dense in this space. Because this space looks a lot like [tex]\mathbb{R}\setminus \mathbb{Q}[/tex], which is completely metrizable, and hence the Baire theorem holds in this case.

Nevertheless, I think the space is not locally compact, but you'll have to show it directly.
Here's a hint. Consider the canonical embedding [tex]i:X\rightarrow \mathbb{R}[/tex]. If K is a compact subset of X, then i(K) is compact in [tex]\mathbb{R}[/tex]. This gives you a good characterization of compact sets...
 

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