A definition of wedge sum can be found here:


<http://en.wikipedia.org/wiki/Wedge_sum>


My professor has claimed that wedge sums of path connected spaces X and Y are well-defined up to homotopy equivalence, independently of choice of base points x0 and y0. Base point here means the points that are identified under the equivalence relation forming the wedge product out of the disjoint union topology of X and Y.


Recall homotopy equivalence of X and Y means that there is f:X->Y and g:Y->X continuous with gf and fg homotopic to the identity.


With these definitions, please prove my professor's claim, which I have failed to do for a week. (It is left as an exercise in his lecture.)


Thanks.



The Wedge Sum of path connected topological spaces
A counterexample is shown on the cover of the paperback edition of the classic textbook Homology Theory by Hilton and Wylie. This can be viewed on the amazon webpage for the book. The example consists of the wedge of two copies of a cone, the cone on the sequence 1/2, 1/3, 1/4, ... together with its limit point 0. With one choice of basepoints the wedge is not contractible, but with other choices it is.


