Thursday, 17 November 2016

general topology - identification of the wedge sum with subset of the cartesian product

Let (X,x0),(Y,y0),(Z,z0) be based spaces.
Define the wedge sum
XYZ to be XYZ modulo x0y0z0.
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:XYZS;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 XYZ to give a homeomorphism. Is this argument correct and especially is f continuous?

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