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?