Finite cartesian product of connected space is connected

In summary: You will choose them so that (a,b) lies on them.In summary, the theorem states that a finite cartesian product of connected spaces is also connected. This is proven using the concept of homeomorphism and the projection onto the individual spaces. The "base point" mentioned in the proof is just an arbitrary point and is not crucial to understanding the proof. The key is choosing the T's in a way that (a,b) lies on them, making the union of the T's connected.
  • #1
princy
14
0
"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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  • #2


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).
 
  • #3


If

[tex]A\times B=U\cup V[/tex]

with U,V nonempty disjunct opens, then

[tex]A=\pi_A(U\cup V)=\pi_A(U)\cup \pi_A(V)[/tex]

or

[tex]B=\pi_B(U\cup V)=\pi_B(U)\cup \pi_B(V)[/tex]

is the union of two non-empty disjoint opens. (Here \pi_C is the projection onto C.)
 
  • #4


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..
 
  • #5


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..
 
  • #6


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...
 
  • #7


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..
 
  • #8


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

What is a finite cartesian product?

A finite cartesian product is an operation on sets that combines elements from two or more sets to create a new set. It is denoted by the symbol "×".

What does it mean for a space to be connected?

A space is connected if it cannot be divided into two non-empty open sets that are disjoint from each other. In other words, there is no way to separate the space into two distinct parts.

What is the definition of a connected space?

A connected space is a topological space in which every pair of points can be joined by a path. This means that there are no "holes" or gaps in the space, and every point is reachable from any other point.

Why is the finite cartesian product of connected spaces connected?

This is a fundamental property of topological spaces. The finite cartesian product of connected spaces is connected because any open sets in the product space can be written as a finite product of open sets in the individual spaces. Since each individual space is connected, the product space cannot be divided into two non-empty open sets that are disjoint from each other.

Can the finite cartesian product of infinitely many connected spaces be connected?

No, in general, the finite cartesian product of infinitely many connected spaces may not be connected. This is because there can be infinitely many ways to combine open sets from each individual space, which can result in a disconnecting open set in the product space.

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