This is problem 2 in Bredon I.14, on homotopy. I need to prove that $X$ = union of the 2-sphere with the unit 2-cell going through the origin is homotopy equivalent to $Y$ = one-point union of two 2-spheres. I tried to consider mapping cones, and made some progress however I'm unsure of some of the steps. Here are the maps I considered:
Define $f: S^1 \rightarrow S^2$ by letting $f(x) = (0,0,1)$ i.e the north pole. Then the mapping cone is homeomorphic to $X$. Second, let $g: S^1 \rightarrow S^2$ be the inclusion map i.e $g(x,y) = (x,y,0)$. Then I think however I'm not completely sure that the mapping cone of $g$ is homeomorphic to $Y$. If this is true, then I'm done with the proof since proving $f$ is homotopic to $g$ is easy. However I'm not completely sure whether the second homeomorphism holds. Can anyone give any insights? Should I even be thinking of mapping cones? I just don't see how constructing an explicit map between those spaces would work.