Connectedness and product of sets

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Homework Statement

If M is connected and N is connected, prove that MXN must be connected.

Homework Equations

A set M is connected if it has no separation. A separation is 2 nonempty open sets A and B such that A union B is M and A intersection B is the empty set.

If X is connected then f(x) is connected

The Attempt at a Solution

I was thinking of constructing a continuous non-constant function f:M--->MxN and then proving MxN being disconnected implied M or N was disconnected. It's a really rough idea but I'm wondering if I'm heading in the right direction? If not can someone tell me what an easier way to prove this would be?

 

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Perhaps try to construct a separation in MxN and see why it is impossible.

 

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Perhaps try to construct a separation in MxN and see why it is impossible.

Yes, that is the main thing I'm trying to do. I just want to know what to do to prove that it is impossible.

 

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Anyone?

 

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My professor gave me the following hint and I think I can figure it out from here. I just want to know why M and N are homeomorphic. It is not in any of the theorems in the book. There is a theorem that says if M is homeomorphic to N and M is connected then N is connected.

"I think that this is the wrong way to approach it. Here's something to think about: MxN has lots of connected subsets -- for example, Mx{y} for any point y in N (since it is homeomorphic to M), and {x}xN, for any x in M (since it is homeomorphic to N). What can you do with them?"