# Is this Space Locally Compact?

• talolard
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.
talolard

## Homework Statement

Is the space $$X=\mathbb{R\backslash}\left\{ a+b\sqrt{2}\::\: a,b\in\mathbb{Q}\right\}$$ 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

Hmm, nice try. But I think that the intersection of dense subsets IS dense in this space. Because this space looks a lot like $$\mathbb{R}\setminus \mathbb{Q}$$, 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 $$i:X\rightarrow \mathbb{R}$$. If K is a compact subset of X, then i(K) is compact in $$\mathbb{R}$$. 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.

• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Topology and Analysis
Replies
5
Views
190
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
545
• Topology and Analysis
Replies
4
Views
337
• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
0
Views
448
• Math POTW for University Students
Replies
1
Views
1K