Closed Sets

Date: 02/27/99 at 07:37:29
From: Michael
Subject: Closed Sets Problem

We showed in class that a set is closed if and only if it contains all 
of its limit points. I want to know how to use this property to show 
that a union of finite closed sets and the intersection of any number 
of closed sets are closed.

Date: 02/27/99 at 07:50:43
From: Doctor Allan
Subject: Re: Closed sets problem

Assume the union of closed sets C_1, C_2,..., C_n is not closed. This 
means that there is a limit point, x,  outside the union. All limit 
points of the union must belong to one of the C_i's (why?), so x must 
belong to one of the C_i's. Does it belong to C_1? No, because if it 
did, it would belong to the union as well. In addition, it does not 
belong to any of the other C_i's, yielding a contradiction. Therefore 
the union must be closed.

Similar reasoning can be used for your question on intersection.

Sincerely,

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