Must the intersection of connected sets be connected?

Question

Must the intersection of two connected sets be connected?

I believe the answer is no, but I am not entirely sure. I think a counter example would be a set that intersects another set in more than one area, yet those intersections are disjoint. (Think of a cylinder that extends through one end of a donut, the hole in the middle, and then the other end of the donut).

Think of two horseshoe-shaped regions. – Daniel Fischer Nov 26 '13 at 23:53 — Daniel Fischer, Nov 26 '13 at 23:53
http://mathworld.wolfram.com/UnicoherentSpace.html $;$ – Nov 27 '13 at 06:43 — , Nov 27 '13 at 06:43

score 28 · Accepted Answer · answered Nov 26 '13 at 23:58

28

No. Consider the intersection of the line segment and the circle in $\varnothing$.

answered Nov 26 '13 at 23:58

Brian M. Scott

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To me, $\not \exists$ a better answer. +1.. – coffeemath Nov 27 '13 at 01:06
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Fantastic answer! – Machine Caliber Apr 13 '15 at 17:49
how can you come up with this counter examples so quick! – ILoveMath Nov 24 '16 at 04:12
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@ILoveMath: Practice and experience, mostly. – Brian M. Scott Nov 24 '16 at 04:47

Must the intersection of connected sets be connected?

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