# Is a Simply Connected Space Homotopically equivalent to a point

If X is a simply connected Space is it homotopically equivalent to a point in the space, I know this holds in $\mathbb{R}^n$ but only because of its algebraic properties does it hold for general topologies?

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A sphere of dimension 2 is simply connected and not equivalent to a point.

A space homotopically equivalent to a point not only has its fundamental group trivial but also all its higher homotopy groups. There is a famous theorem of Whitehead that says that, provided the space is sufficiently good, having all homotopy groups trivial does imply contractibility; sufficiently good means, for example, being a CW-complex.

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Ah thanks a lot for that, evidently being a bit slow –  user147606 May 3 '14 at 17:23