Problem about a connected subspace

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In summary: Thank you, now I see the mistake in the counterexample I thought I had.in fact this is almost the same as the proof that a product of connected spaces is connected.
  • #1
facenian
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Helo, I believe that the folowing exercise from Topology by Munkres is incorrect:
"Let A be a proper subset of X, and let B be a proper subsert of Y. If X and Y are conected, show that
##(X\times Y)-(A\times B)## is connected"
I think I can prove it wrong however I'm not sure and would like to discuss it.
 
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  • #2
facenian said:
Helo, I believe that the folowing exercise from Topology by Munkres is incorrect:
"Let A be a proper subset of X, and let B be a proper subsert of Y. If X and Y are conected, show that
##(X\times Y)-(A\times B)## is connected"
I think I can prove it wrong however I'm not sure and would like to discuss it.

Then give a counterexample.
 
  • #3
Math_QED said:
Then give a counterexample.
I think I have it, however I don't want to mislead someone else to a probably mistaken answer that's why I would like to know if anybody already now the solution or has an opinion.
I think I must be wrong otherwise there is a mistake in Munkres' book
 
  • #4
Let's just look in the (x,y) plane R^2, and remove the product (0,1) x (0,1). Do you see how to connect up all points of the complement by drawing vertical and horizontal lines? The same thing works in general. try it yourself after removing from the plane say all points having both coordinates rational. I.e. remove QxQ from RxR. The point is in general that in the product space XxY, all subsets of the form {a} x Y and Xx{b} are connected, (and meet at (a,b)), and the union of two connected sets that meet is also connected. Does that do it? I have not written anything down.
 
  • #5
mathwonk said:
Let's just look in the (x,y) plane R^2, and remove the product (0,1) x (0,1). Do you see how to connect up all points of the complement by drawing vertical and horizontal lines? The same thing works in general. try it yourself after removing from the plane say all points having both coordinates rational. I.e. remove QxQ from RxR. The point is in general that in the product space XxY, all subsets of the form {a} x Y and Xx{b} are connected, (and meet at (a,b)), and the union of two connected sets that meet is also connected. Does that do it? I have not written anything down.

Thank you, now a see the mistake in the counterexample I thought I had.
 
  • #6
in fact this is almost the same as the proof that a product of connected spaces is connected.
 

1. What is a connected subspace?

A connected subspace is a subset of a larger topological space that cannot be divided into two or more non-empty open subsets. This means that all points in the subspace are "connected" in some way, and there are no gaps or breaks in the space.

2. How is a connected subspace different from a disconnected one?

A disconnected subspace is a subset of a topological space that can be divided into two or more non-empty open subsets. This means that there are gaps or breaks in the space, where points in one subset cannot be connected to points in another subset.

3. Why is it important to study connected subspaces?

Studying connected subspaces is important because they have unique properties and can help us understand the larger topological space. For example, connected subspaces play a crucial role in the study of continuity and path-connectedness in topology.

4. How can we determine if a subspace is connected?

There are a few different methods for determining if a subspace is connected. One way is to show that the subspace cannot be divided into two or more non-empty open subsets. Another way is to show that any two points in the subspace can be connected by a continuous path.

5. Can a connected subspace be disconnected?

No, by definition, a connected subspace cannot be disconnected. If a connected subspace were to be divided into two or more non-empty open subsets, it would no longer be considered connected. However, a disconnected subspace can potentially become connected if it is further divided into smaller subsets.

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