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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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    @Aloizio Indeed, this is only true in the plane.2018-11-01

1 Answers 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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    @Ragib: Oh, well. I’ve never been able to resist a pun.2012-07-20