Finite cartesian product of connected space is connected

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The discussion revolves around understanding the theorem that states the finite Cartesian product of connected spaces is connected. Participants express confusion about the proof, particularly regarding the concepts of base points and homeomorphism. It is clarified that a base point is an arbitrary point that serves as a common element in the proof, which is not crucial for understanding the overall argument. The proof involves showing that the union of certain sets, which include the base point, remains connected. Ultimately, the conversation emphasizes the importance of grasping these foundational concepts to comprehend the theorem fully.
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"finite cartesian product of connected space is connected"

hi am not able understand the theorem that.. "finite cartesian product of connected space is connected".. what is a base point? how it is related to homeomorphism? can anyone explian?
 
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It would help if you could link to a proof that you have tried to study, and point out the step you're having difficulties with. (See if you can find the relevant page at Google Books).
 


If

A\times B=U\cup V

with U,V nonempty disjunct opens, then

A=\pi_A(U\cup V)=\pi_A(U)\cup \pi_A(V)

or

B=\pi_B(U\cup V)=\pi_B(U)\cup \pi_B(V)

is the union of two non-empty disjoint opens. (Here \pi_C is the projection onto C.)
 


Fredrik said:
It would help if you could link to a proof that you have tried to study, and point out the step you're having difficulties with. (See if you can find the relevant page at Google Books).


no am not able get any idea in that proof..in james.r.munkres book it is given that..
i can only get the idea that it is proved by the use of homeomorphism.. I am not getting the idea how it is related to that figure given.. and the base point.. i think there is something very basic which i need to get..
 


no am not able get any idea in that proof..in james.r.munkres book it is given that..
i can only get the idea that it is proved by the use of homeomorphism.. I am not getting the idea how it is related to that figure given.. and the base point.. i think there is something very basic which i need to get..
 


What Munkres wants to do is apply theorem 23.3. The point in common in that theorem is what he calls the "base point". The base point is just an arbitrary point that will become the "point in common". If you don't get it, then just forget that he mentioned base point. It's not at all important for the proof...
 


ok so base point is just an arbitrary point.. it says that the union of all the Ts are connected because it has (a,b) as the common point.. how come (a,b) common to all the Ts.? can u explain it.?am not getting it..
 


Because you choose the T's exactly so that (a,b) lies on them. How did you choose the T's?
 

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