Is the Set of Rational Numbers with the Relative Topology Not Locally Compact?

[bighadi]

Homework Statement

Prove that the set of rational numbers with the relative topology as a subset of the real numbers is not locally compact

Homework Equations

none

The Attempt at a Solution

I am totally confused and want someone to give me a proof. I have looked at some stuff online but nothing made sense.

 

[statdad]

You would like someone to give you a proof? I would like someone to give me a Saleen S7. Neither is going to happen in the immediate future. Note this from the guidelines.

1) Did you show your work? Homework helpers will not assist with any questions until you've shown your own effort on the problem. Remember, we help with homework, we don't do your homework.

 

[bighadi]

Okay, let's start with a definition:

A topological space X is locally compact if for each p \in X, there is an open set W such that p \in W and \overline{W}, the closure of W, is compact.

So, since we are trying to show that Q is not locally compact, we need to show that there exists p \in X such that for all open sets W and p \notin W and \overline{W} is not compact.

Assuming I wrote the negation of being locally compact correctly, how do I show the the above statement?

[By the way, I would also love getting a Saleen S7.]

I think I posted this under the wrong section. I meant to put it under "Calculus and beyond".