Connected/Disconnected all the same to me

[marlen]

Connected/Disconnected...all the same to me...

My question for all of you ladies and gentlemen is

what would be considered as an example of a connected set in R squared that becomes disconnected when we remove one point.

My answer would be sin(x/2), but is there a simpler example.

 

[quasar987]

consider two closed balls with just one point in common.

Or like you said, a curve that does close in on itself.

 

[marlen]

consider two closed balls with just one point in common.

Or like you said, a curve that does close in on itself.

Huh? I'm confused.

Would 1/x work? What would be a solid example, one that I could understand?

 

[marlen]

S = f(x; y) : y = 1/x; 0 < x  1g [ f(x; 0) : 1  x  0g would this work

 

[marlen]

S = f(x; y) : y = 1/x; 0 < x  1g [ f(x; 0) : 1  x  0g would this work

TOTALLY DISREGARD THIS COMMENT

 

[marlen]

S = {(x,y): y = 1/x, 0 < x $$\leq$$ 1} $$\cup$$ {(x,0): -1 $$\leq$$ x $$\leq$$ 0}

I meant would this work?

 

[Dick]

S = {(x,y): y = 1/x, 0 < x $$\leq$$ 1} $$\cup$$ {(x,0): -1 $$\leq$$ x $$\leq$$ 0}

I meant would this work?

No, because S isn't connected. Either part of S would work.

 

[HallsofIvy]

Or A= {(x,y)| y= 0} will do.