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Let $(X, \tau)$ be a topological space. Let $U,V$ be two simply connected sets.

Prove or disprove: $U \cap V$ is simply connected.

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HINT: Find two simply connected subsets of the plane whose intersection isn’t even connected; a pair of kidneys will work, if you orient them properly. ;-) –  Brian M. Scott Jul 20 '12 at 22:52
It is perhaps worth noting that this is essentially the only way in which $U\cap V$ can fail to be simply connected; if $U\cap V$ is path-connected then it is simply connected by Mayer-Vietoris. –  Alex Becker Jul 20 '12 at 22:58
Thanks, although it sounds like you're killing a fly with a steamroller... –  pre-kidney Jul 20 '12 at 23:03
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1 Answer

up vote 7 down vote accepted

Let $S^1$ be the circle in $\mathbb R^2$, $U=\{(x,y)\in S^1: x\geq 0\}$ and $V=\{(x,y)\in S^1: x\leq 0\}$. Then $U$ is the right half of a circle and $V$ is the left half, both of which are simply connected. What is $U\cap V$?

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You've got some skinny kidneys :) –  pre-kidney Jul 20 '12 at 23:00
I was about to post left and right hemispheres intersecting on a circle, but this is even simpler! +1. –  Ragib Zaman Jul 20 '12 at 23:03
I was tempted to post a one-character hint: $\between$ –  Brian M. Scott Jul 20 '12 at 23:04
But then you couldn't make your kidney pun... –  pre-kidney Jul 20 '12 at 23:06
@BrianM.Scott I like it! But that is probably (just a little) more work to describe formally than this. –  Alex Becker Jul 20 '12 at 23:06
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