Let $$X=S^1\cup \{0\}\times [-1,1]\ \text{and } Y=\{S^1+(1,0)\}\cup \{S^1+(-1,0)\}.$$ Are these two spaces homotopic?
I can see it in the picture that they are homotopic. As in the line if we identify $(0,1)$ and $(0,-1)$ then we will get figure 8 which is homotopic to $Y$. But how do I show it mathematically?
