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## Homework Statement

Let X be an infinite set with the cofinite topology. Show that the product topology on XxX (X cross X) is strictly finer than the cofinite topology on XxX.

## Homework Equations

None

## The Attempt at a Solution

So we know that a set U in X is open if X-U is finite. Hence, U contains infinitely many points of X except for a finite subcollection of points, i.e. x_i for i=1, ..., n. Thus, under the product topology, let UxV is open in XxX where U and V are open in X. We can then get our closed sets from UxV as {x_1, ..., x_n}x{y_1, ..., y_n}. Under the cofinite topology, XxX produces an open set M s.t. XxX-M={m_1, ..., m_n}. I argue that we get more closed sets under the product topology than under the cofinite topology, although I'm having trouble showing this rigorously.

If anyone has a better method or would like to fill in the dots of my method, I would appreciate your input.