Metric Spaces

Date: 12/11/2002 at 00:43:16
From: Carmen McCullough
Subject: Topology (metric spaces)

Let (X,d) be a metric space with the discrete metric. 
Prove that no subset of X has a limit point.

I've already proved that every subset of X is closed and every subset 
of X is open, but if every subset of X is closed each subset contains 
its limit points, while if open maybe maybe not. Since every subset is 
open and closed, then is that why there are no limit points? I can't 
figure out how to explain this one.

Date: 12/11/2002 at 01:30:20
From: Doctor Mike
Subject: Re: Topology (metric spaces)
  
Hello Carmen,
  
This is one of those problems that is so easy it is difficult.
  
The best strategy in such situations is to carefully read the 
definition of the property involved, in this case Limit Point.
  
I took my own advice and pulled my copy of _Set Theory and Metric 
Spaces_ by Irving Kaplansky, down off the shelf, to read that a limit 
point of a set A is a point x that lies in the closure of A-{x}. Other 
books would have similar, and I would hope equivalent, definitions.  
  
I ask you this: If you have proved that EVERY set is closed, what is 
the "closure of A-{x}" mentioned in the definition?
  
Here is something else to think about: If {x} is an open set (which 
it is) and if A is any subset of the metric space, how many points of 
A-{x} can the open set {x} contain?
  
One final thing: Describe the "open ball" consisting of all points in 
the metric space at a distance LESS THAN 1.0 from a specific point x.  
That is, what is { p | d(x,p) < 1.0 }?
  
These are three good hints.  Go forth and prove this thing!

- Doctor Mike, The Math Forum
  http://mathforum.org/dr.math/