I Problem about a connected subspace

[facenian]

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.

 

[member 587159]

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.

 

[facenian]

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

 

[mathwonk]

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.

 

[facenian]

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.

 

[mathwonk]

in fact this is almost the same as the proof that a product of connected spaces is connected.