# Is this Locally Compact?

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

## Answers and Replies

Staff Emeritus
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.
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...