How to show that these two spaces are homotopy equivalent?

Question

If $X_{1}$ = $\mathbb{S}^{2} \vee S^{1} $( here $S^{1}$ is attached outside to $\mathbb{S}^{2}$ ) and $X_{2}$ = $\mathbb{S}^{2}$ with a diameter joining north pole and south pole. In the second case it is becoming wedge but $S^1$ is inside of $\mathbb{S}^{2}$. It is not clear to me how this two space are homotopy equivalent. How to continuously deform one space to another?

Answer 1

A figure for this situation is

which is Fig 7.9 taken from the pdf of Topology and Groupoids.

Contracting to a point the outside line on the right gives the left hand figure. On the other hand just moving the the top dot of the right hand figure over the sphere to coincide with the bottom dot, gives $S^2 \vee S^1$. (I may have said this somewhere else on this site!)