I read on wiki* that a (pointed) topological space is simply connected iff its fundamental group is trivial. But I don't see how this in accordance with the R² caracterisation that U is simply connected iff it is path-connected and has no holes in it.(adsbygoogle = window.adsbygoogle || []).push({});

Take the closed unit-disk with a point of its boundary as base point for example. Then what are the elements of the fundamental group? There is the trivial loop, which is the identity element, and then there are the homotopy equivalence classes of the loops that circles the boundary S^1 once, twice, three times, etc. such that the fundamental group is isomorphic to Z, not {e}.

*http://en.wikipedia.org/wiki/Simply_connected#Formal_definition_and_equivalent_formulations

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# Topology: simple connectedness and fundamental groups

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