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Every compact contractible planar set is simply connected. Does the converse hold within the class of path-connected, locally path-connected planar continua X? (Here, as usual, simply connected means that any closed curve is null-homotopic). So, since the space X above is itself a closed curve (Hahn-Marzurkiewiec), one could imagine that the contraction of this curve to a point could yield the contraction of the space to a point (=contractibility). But is this true? (I guess not, since I do not know of a property that distinguishes planar Peano curves from those in higher dimensions, and there it is definitely false (sphere S_2)). But a counterexample would be nice. Thanks. PS I asked part of this question on another forum; without response).

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  • $\begingroup$ Higher homotopy groups are known to vanish for planar sets (this is nontrivial). Maybe there is a way to argue that in your setting the subset has the homotopy type of a CW complex. $\endgroup$ – Moishe Kohan Oct 6 '15 at 1:01

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