Let (X,x0),(Y,y0),(Z,z0) be based spaces.
Define the wedge sum
X∨Y∨Z to be X⊔Y⊔Z modulo x0∼y0∼z0.
How does this wedge sum relates to the subspace S={(x,y,z)∈X×Y×Z|only one of the entries is not the base point}
I Know they are identified but i'm not sure how to prove it. Let
f:X⊔Y⊔Z→S;f(x)=(x,y0,z0),f(y)=(x0,y,z0),f(z)=(x0,y0,z)
Then obviously f is surjective and such that
f(x0)=f(y0)=f(z0) hence f facotors through X∨Y∨Z to give a homeomorphism. Is this argument correct and especially is f continuous?
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