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Cofinite topology vs. Product Topology

  • #1

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.
 

Answers and Replies

  • #2
428
1
I'm not sure you described all the closed sets in the product topology correctly. Try taking U to be everything but zero in the integers, think you'll find the complement of UxU is infinite.

That should get you well on your way, at least for the topologies not equal part.
 
  • #3
I'm not sure you described all the closed sets in the product topology correctly. Try taking U to be everything but zero in the integers, think you'll find the complement of UxU is infinite.

That should get you well on your way, at least for the topologies not equal part.
I realized this a few hours after posting, so your response reinsured what I had come up with. Thank you. Drawing a picture of XxX with X=reals made this very clear. It's amazing how much drawing a simply illustration can help.
 

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