Notes on Products

Definition:

Given topological spaces and , a basis for the product topology on is given by all sets of the form , where , and are open sets.

Note that the sets form a basis since so the sets are in fact closed under finite intersections.

Remarks:

Given topological spaces and , , let and be the projection functions (i.e. ) The product topology on can also be described as the topology generated by all sets of the form , or where , and are open sets. By generated we mean "take arbitrary unions of finite intersections of such sets."

Note that:

1. 

2. and are continuous with respect to this topology, essentially by definition.
   
   

3. is continuous if and only if and are continuous.

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Homework:

Due: Thursday 20

• Given a topological space , a point set , and a function , show that the collection of sets

  , any open set forms a topology on such that is continuous. (see the definition of a Quotient Space)

• Given two topological spaces and , a point set , and two function and , show that the collection of sets , any open sets are a basis for a topology on such that is continuous .

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Lemma:

Let then is continuous, where . Similarly for

Proof:

We need only show that the inverse image of basis elements are open. But or depending on whether or not

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Theorem:

If and be two compact topological spaces then is compact.

Proof:

In general, since any open set is the union of basis elements it suffices to show that any open covering by basis elements admits a finite subcovering. (WHY?)

Let be an open covering of Since is compact and is continuous, for each , we can select a finite subset such that covers .

Let .

Note:

Next, since is compact, there is a finite subset of the that covers . Thus

covers , as does since for each .