Vito Prosciutto: Teaching community college math on the road to a PhD.

Wednesday, June 15, 2005

Topology question (does this question make sense?) 

Munkres, p.92, question 5:

Let X and X' denote a single set in the topologies T and T' respectively. Let Y and Y' denote a single set in the topologies U and U' respectively. Assume that these sets are nonempty.

(a) show that if T' is a superset of T and U' is a superset of U, then the product topology in X'xY' is finer than the product topology on XxY.

So here's my question: Are we to assume that the X and X' differ only in the topologies that they belong to? i.e., that we might be talking about the open set (0,1) in R and RK? Because otherwise, if they're two arbitrary sets in the topologies then there's no reason to assume that the subspace topologies of the product topology are even comparable. That's how I'm interpreting the question and it seems that the statement is true and easily provable, but I'd like to get some confirmation on this.

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