There are some facts that are really impressive after you remove a point to a space. Some typical examples are the existence of exotic spheres or the fact that $S^4$ is not almost complex. Or some not differentiable manifolds that become differentiable manifold after removing a point.
closed as unclear what you're asking by YCor, Ian Morris, abx, Pace Nielsen, Alexandre Eremenko Jun 19 '18 at 20:44
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There is this fun fact in complex geometry: a holomorphic vector bundle over non-compact Riemann surface is necessarily trivial. This is obviously not true for closed Riemann surfaces, since there are topological obstructions such as Chern classes, and there are 'holomorphic' obstructions for for holomorphic vector bundles admitting continuous trivialization (what does Jacobian parametrize?).