(Fulton's Algebraic Topology: A First Course, Problem 6.28) Can we find a closed set $X\subseteq\mathbb R^2$, such that there's a cycle $\gamma$ in $X$ not homologous to zero but for each $P\in\mathbb R^2\setminus X$, the winding number $n(\gamma,P)=0$?


It seems that $X$ should be weird. Apparently, a counterexample $X$ shouldn't be totally disconnected. I have tried something like modified comb space, but didn't get any idea on constructing a counterexample.

PS: There's no counterexample when $X\subseteq\mathbb R^2$ is open.

Any idea? Thanks!


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