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Let $U,V$ be two simply connected subsets of a topological space.

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

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    $\begingroup$ 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. ;-) $\endgroup$ Jul 20, 2012 at 22:52
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    $\begingroup$ 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. $\endgroup$ Jul 20, 2012 at 22:58
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    $\begingroup$ Thanks, although it sounds like you're killing a fly with a steamroller... $\endgroup$
    – pre-kidney
    Jul 20, 2012 at 23:03
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    $\begingroup$ @AlexBecker That is not true. Consider $U$ as the upper closed hemisphere of $S^2$ and $V$ as the lower one. $\endgroup$
    – Aloizio Macedo
    Mar 14, 2017 at 0:41
  • $\begingroup$ @Aloizio Indeed, this is only true in the plane. $\endgroup$
    – domotorp
    Nov 1, 2018 at 15:27

1 Answer 1

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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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    $\begingroup$ You've got some skinny kidneys :) $\endgroup$
    – pre-kidney
    Jul 20, 2012 at 23:00
  • $\begingroup$ I was about to post left and right hemispheres intersecting on a circle, but this is even simpler! +1. $\endgroup$ Jul 20, 2012 at 23:03
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    $\begingroup$ I was tempted to post a one-character hint: $\between$ $\endgroup$ Jul 20, 2012 at 23:04
  • $\begingroup$ But then you couldn't make your kidney pun... $\endgroup$
    – pre-kidney
    Jul 20, 2012 at 23:06
  • $\begingroup$ @BrianM.Scott I like it! But that is probably (just a little) more work to describe formally than this. $\endgroup$ Jul 20, 2012 at 23:06

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