# Metric Spaces

• Sep 21st 2009, 01:29 PM
tedii
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?
• Sep 21st 2009, 01:38 PM
Plato
• Sep 21st 2009, 03:51 PM
tedii
Quote:

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?
• Sep 21st 2009, 05:03 PM
Plato
Quote:

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 $D(a,b)>1???$
• Sep 21st 2009, 06:02 PM
tedii
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?