Let $X$ be an open Riemann surface and $K$ an open, relatively compact, connected subset of $X$. It seems to me that it is impossible for $\overline{K}$ to strongly deformation retract onto its boundary $\partial K$. (I feel that you'd have to rip it.) But if I'm right, I don't know how to convince myself of that rigorously. Or perhaps I'm wrong?
2026-02-23 04:49:46.1771822186
A compact Riemann surface with boundary strongly deformation retracting onto its boundary
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