Tag Archives: Tychonoff Theoream

Worksheet for the finite product Tychonoff Theorem.

Leave a comment Posted by on March 30, 2015

Let X \times Y be the product of two compact spaces X and Y . We want to show that X \times Y is compact. Let \mathscr{U} be an open cover of X \times Y ; we can assume that each open set in this cover is of the form U_{\beta} \times V_{\beta} where U_{\beta}, V_{\beta} are open sets in X, Y respectively (remember that the box and product topologies are equivalent when the product is finite).

To understand this theorem, prove each of the following:

1. Let x \in X . Then \{x \} \times Y is homeomorphic to Y . (the homeomorphism is f(y) = (x, y) \in X \times Y

2. \{x \} \times Y is a compact subset of X \times Y .

3. A finite subcollection of \mathscr{U} covers \{x \} \times Y ; call this \cup^{k_x}_{i=1} U_{i,x} \times V_{i,x} . The x in the subscript and in the number of open sets indicates that this open subcover is associated with \{x \} \times Y .

4. \cap^{k_x}_{i=1}U_{i,x} is an open set in X which contains x .

5. The open set U_{i,x} \times Y is a subset of the open set \cup^{k_x}_{i=1} U_{i,x} \times V_{i,x} . We say that the latter open set is the “part of the open cover” associated with U_{i,x}. 6. Do this procedure for each latex x \in X $. Then if U_x = \cap^{k_x}_{i=1}U_{i,x} , then \cup_{x \in X}U_x is an open cover for X . Therefore there is a finite subcover U_{x_i}, i \in \{1, ...m \} which covers X .

7. Then the collection U_{x_i} \times Y, i \in \{1, ...,m \} covers all of X \times Y . (show that if (x,y) \in X \times Y then (x,y) lies in at least one of these open sets.

8. The desired “finite open subcover of X \times Y is the collection of associated finite subcovers discussed in 5.

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