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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