# Different ways to wedge spaces

I am not a topologist, so please excuse me if the question is trivial.

Suppose I am given three nice, path-connected spaces. Then I can think of two ways to wedge these spaces: join all three at a common basepoint, or have one space in the "middle" with two basepoints (so I guess there are four ways in total). Do these constructions result in the same space up to homotopy? My guess would be to take a path between the two basepoints in the "middle" space and contract it to a point, but I understand that it is sometimes a subtle matter to tell if this sort of thing is a homotopy equivalence.

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This is already an issue for two spaces, right? It's not clear that the wedge doesn't depend on the choice of basepoint in each space. – Qiaochu Yuan Jul 30 '12 at 20:06

What you need is that your input spaces are path connected and moreover, given your two choices of base-point you want a homotopy-equivalence $(X,x_0) \simeq (X,x_1)$.
If $X$ is also locally contractible, then $(X, x_0) \simeq (X, x_1)$. But this is probably too strong a condition for the homotopy equivalence. – Shaun Ault Jul 30 '12 at 21:38