Is this Space Locally Compact?

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In summary, the conversation discusses whether the space X=\mathbb{R\backslash}\left\{ a+b\sqrt{2}\::\: a,b\in\mathbb{Q}\right\} is locally compact and how to show a violation of the Baire theorem in this space. The conversation also mentions the use of the canonical embedding i:X\rightarrow \mathbb{R} to characterize compact sets in X.
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talolard
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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
 
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  • #2
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...
 

1. What does it mean for a space to be locally compact?

Locally compactness is a property of a topological space where every point in the space has a compact neighborhood. This means that for every point in the space, there exists a compact subset that contains that point and is contained within an open set.

2. How is locally compactness different from compactness?

Compactness is a global property of a space, meaning that the entire space is compact. Locally compactness, on the other hand, is a local property, meaning that it only has to hold for each individual point in the space and its surrounding neighborhood.

3. Are all locally compact spaces also compact?

No, not all locally compact spaces are compact. A locally compact space may have infinitely many points, meaning that it cannot be compact. Compactness requires a space to be finite, whereas local compactness does not have this restriction.

4. Is the real line locally compact?

Yes, the real line is locally compact. Every point on the real line has a compact neighborhood, such as a closed interval, that contains that point and is contained within an open set.

5. Can a non-Hausdorff space be locally compact?

Yes, a non-Hausdorff space can still be locally compact. The requirement for local compactness is that each point has a compact neighborhood, while Hausdorffness is a stronger condition that requires every two distinct points to have disjoint neighborhoods.

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