Let $X=\mathbb{S}^{2}/\mathbb{S}^{0}$ be the space resulting from collapsing one pair of antipodal points on a two-dimensional sphere to a single point. Let $Y$ be $\mathbb{S}^{2} \vee \mathbb{S}^{1}$. Specifically, $Y$ fits in the following pushout diagram. $\require{AMScd}$ \begin{CD} \ \{a\} @>{f}>> \mathbb{S}^{2}\\ @VgVV @VVV\\ \ \mathbb{S}^{1} @>>> Y \end{CD}
Where $\{a\}$ is a one-point space with $f(a)=(1,0,0)$ and $g(a)=(1,0)$.
Are $X$ and $Y$ homotopy equivalent? If they are, how can I see it? Can it be solved using the fact that a space $X$ has the homotopy extension property with respect to a subspace $A$ if and only if $([0,1]\times A) \cup(\{0\}\times X)$ is a retract of $[0,1]\times X$?