(adsbygoogle = window.adsbygoogle || []).push({}); If [itex]X[/itex] and [itex]Y[/itex] are two connected topological spaces then so is [itex]X \otimes Y[/itex].

I want to understand the proof of this theorem but I am having some difficulties. Even though we went over it in class, it is still unclear to me.

The professor constructed this continuous function:

[tex]f:X\otimes Y \rightarrow \{0,1\}[/tex]

Where [itex]\{0,1\}[/itex] is a discrete topological space. Then he shows that [itex]f[/itex] is constant. He then claimed that [itex]\{x\} \otimes Y[/itex] is homeomorphic with [itex]Y[/itex] hence this subspace ([itex]\{x\}\otimes Y[/itex]) is connected - since [itex]Y[/itex] is.

Now this does not make sense to me and I wouldn't be suprised if it didn't make sense to any of you. If you think you have a better way of explaining the proof (it doesn't have to be this one) then I would appreciate the effort.

Im not sure exactly why one would begin by setting up a continuous function which maps points in the product space [itex]X \otimes Y[/itex] to either 0 or 1 in the discrete topological space, then claim that the function constant - then go on to show that [itex]\{x\} \otimes Y[/itex] is homeomorphic to [itex]Y[/itex].??

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# Product Spaces

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