Given a topological manifold of dimension 2 or more and points [ilmath]p_1[/ilmath], [ilmath]p_1[/ilmath] and [ilmath]q[/ilmath] where [ilmath]q[/ilmath] is neither [ilmath]p_1[/ilmath] nor [ilmath]p_2[/ilmath] then a path from [ilmath]p_1[/ilmath] to [ilmath]p_2[/ilmath] is path-homotopic to a path that doesn't go through [ilmath]q[/ilmath]

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TODO: I'd really like a picture of an open ball, centred at [ilmath]q[/ilmath] and a path going through it, and another picture of the path but moved in the open ball to avoid [ilmath]q[/ilmath]


For a topological manifold, [ilmath]M[/ilmath], of dimension no more than [ilmath]2[/ilmath], points [ilmath]p_1,p_2,q\in M[/ilmath] such that [ilmath]q\ne p_1[/ilmath] and [ilmath]q\ne p_2[/ilmath] and a path, [ilmath]f:I\rightarrow M[/ilmath] (where [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath] - the unit interval) from [ilmath]p_1[/ilmath] to [ilmath]p_2[/ilmath], is path-homotopic to a path that does not go through [ilmath]q[/ilmath][1].


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  1. Introduction to Topological Manifolds - John M. Lee