Proving a Topology

Date: 10/11/98 at 19:41:18
From: Robert
Subject: Topology

I was given this question in class and am trying to understand the 
question. If you could clarify the question I would appreciate it.  
You don't necessarily have to answer the question for me.  

Question.

Let X be an uncountable set of points, and let T consist of the empty 
set and all subsets of X whose complement is finite. Prove that T is 
a topology of X.

I know that in order to show that T is a topology of X, I must show 
that:

   1. The empty set and X are in T (empty set and X are open)
   2. X is closed under finite intersection 
   3. X is closed under arbitrary union

The first part is easy to see since by the definition the empty set is 
in T and all the subsets of X make up T so all of X is in T. I am 
confused about the other two parts though. I will let A be a subset of 
X. My question is where does the complement part come in? This is where 
all my questions about this problem stem. Can you rephrase the question 
to make it more clear?

Thank you in advance for all your help. I really appreciate it.

Robert

Date: 10/13/98 at 18:56:41
From: Doctor Mike
Subject: Re: Topology

Hi Robert,

First note that a better way to say why X is in T would be to give a 
reason why the "complement of X" is finite. What actually is the 
complement of X? Since we are talking about complements in X, the 
answer is the empty set. How many elements are in the empty set?  
Zero. Is zero finite? Yes. There you go. The complement of X in X 
contains the finite number 0 of elements, so X itself is in the 
topology T. 
  
For A to be open, its complement X-A (everything in X which is not in 
A) must be a finite set. You have to show that any time you intersect 
a finite number of open sets, the result is also an open set. Let's 
start with something easier, namely two open sets A and B. Since they 
are open, X-A and X-B are both finite sets. Let's write A+B for the 
intersection of A and B. You have to prove that A+B is open. To do that 
you have to prove that the complement of A+B is finite. In symbols 
X-(A+B) must be finite. Do you know that X-(A+B) is the same as 
(X-A) U (X-B) (where U means Union)? Now it is easy. Because X-A and 
X-B are both finite, the union of these 2 sets must also be finite. Now 
see if you can extend this reasoning from 2 sets to a finite number of 
sets. 
   
For the third part, you have to assume that you have a collection of 
open sets, but not assume that there is only a finite number of them, 
as you could above. You must show that the union of all of these open 
sets is also open. That is, you must show that the union of all of 
these open sets has a finite complement in X.  
  
- Doctor Mike, The Math Forum
  http://mathforum.org/dr.math/