Page 199 - DMTH503_TOPOLOGY

Unit 22: The Tychonoff Theorem

Now D ...  D belongs to D, by (a); therefore this intersection is non-empty, by Notes
1 n
hypothesis.
Theorem 1: (Tychonoff theorem): An arbitrary product of compact spaces is compact is the product
topology:
Proof: Let

X   X ,

  J
where each space X is compact. Let  be a collection of subsets of X having the finite intersection

property. We prove that the intersection

A
A
is non-empty. Compactness of X follows:

Applying Lemma 1, choose a collection  of subsets of X such that   and  is maximal with
respect to the finite intersection property. It will suffice to show that the intersection  D D is
non-empty.
Given J, let  : X X be the projection map, as usual. Consider the collection
 
{ (D)| D }

of subset of X . This collection has the finite intersection property because  does. By compactness

of X , we can for each  choose a point x of X such that
  
x   (D).
 
D
Let x be the point (x )  J of X. We shall show that for x  D for every D ; then our proof will
 
be finished.
First we show that if  – 1 ( ) is any sub-basis element (for the product topology on X) containing
 
–1
x, then  ( ) intersects every element of . The set is a neighbourhood of x in X .Since
    
x  (D) by definition, intersects  (D) in some point  (y), where y D. Then it follows
    
–1
that y  ( )  D.
 
It follows from (b) of Lemma 2, that every sub-basis element containing x belongs to D. And
then it follows (a) of the same lemma that every basis element containing x belongs to . Since
 has the finite intersection property, this means that every basis element containing x intersects
every element of ; hence x D for every D  as desired.
22.2 Summary

 Let X be a set and  a family of subsets of X. Then is said to have the finite intersection
property if for any finite number F , F , ... F of members of . F  F  ...  F .
1 2 n 1 2 n
 Let (X, T) be a topology space. Then (X, T) is compact iff every family  of closed subsets of
X with the finite intersection property satisfies  F .
F 
 An arbitrary product of compact spaces is compact in the product topology.

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