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radou

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**1. Homework Statement**

This seems suspiciously easy, so I'd like to check my reasoning.

**3. The Attempt at a Solution**

I used the following theorem:

If X is a Hausdorff space, then X is locally compact iff given x in X, and a neighborhood U of x, there exists a neighborhood V of x such that Cl(V) is compact and Cl(V) is contained in U.

The rationals are clearly Hausdorff. Assume they are locally compact, so the implication from the theorem holds. But then there are irrational numbers in Cl(V), and such a set cannot possibly be contained in U.