# Math Help - Metric Spaces

1. ## Metric Spaces

Let X be an infinite set. For $p\in X,q\in X$ define,

$d(p,q)=\left\{\begin{array}{cc}1(p\not=q)\\0(p=q)\ end{array}\right.$

Prove that this is a metric. Which subsets of the resulting metric space are open? Which are closed? Which are compact?

Proving it's metric is easy no problem there.

The rest I just want to make sure I'm thinking of this correctly.

Every subset of the metric space is finite, therefore every set is closed, compact, and not open.

Does this sound right?

2. Originally Posted by Plato
On that page it says that $\{x_0\}\mbox{ if }0.

What about when r=1 this still does not contain X? Why is that?

3. Originally Posted by tedii
On that page it says that $\{x_0\}\mbox{ if }0.
What about when r=1 this still does not contain X? Why is that?
Can there be $a~\&~b$ such that $D(a,b)>1???$

4. No, but r can be and it's just at length r=1 that I don't understand why that doesn't contain X. Is it because it's an open ball?