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Define h: A x C -> B x D by the equation

h(a,c)=(f(a),g(c)).

Show h is continuous.

A few weeks ago I completed this exercise. Now, I am working on a problem that would be almost too easy if the converse of the above claim were true. I had trouble trying to construct a counterexample; so I tried to prove it.

Suppose h is continous. Let U be open in B. Then U x D is open in B x D, and by our assumption, h

^{-1}(U x D) is open in A x C. Since h

^{-1}(U x D) = f

^{-1}(U) x g

^{-1}(D),

__f__

^{-1}(U) is open and f is continuous.Does f

^{-1}(U) x g

^{-1}(D) being open in the product space imply f

^{-1}(U) is open in A?