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Exercise 0.21 of Hatcher's Algebraic Topology reads:

If $X$ is a connected Hausdorff space that is the union of a finite number of $2$-spheres, any two of which intersect in at most one point, show that $X$ is homotopy equivalent to a wedge sum of $S^1$'s and $S^2$'s.

I believe I came up with a solution to this, but nowhere did I use the assumption of "Hausdorff". Is this really a necessary assumption? Where would you use the $T_2$ condition in a proof like this? It seems that a union of $2$-spheres would have to be Hausdorff . . .

What am I missing?

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    Could you exhibit a homotopy of maps on non-Hausdorff spaces?2011-10-06
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    What's your proof?2011-10-06
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    A real quick summary: "Lengthen" the intersections into line segments. On each sphere contract all the points where line segments intersect it into one point, so that if the sphere originally intersected $k$ other spheres, there will now be $k$ line segments emanating out of one point. We now essentially have a graph in which the vertices are $2$-spheres. Contract the line segments that make up the graph for as long as you can until there are no other contractible segments left. The result will be a wedge sum of $1$-spheres and $2$-spheres.2011-10-06

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