**Compute the homology groups of the quotient of $\mathbb{S}^2$
obtained by identifying north and south poles to a point.**

I have already calculated the homology groups in one way, but I would like to calculate the homology groups also using the mayer vietoris sequence, so I am trying to find two open $U\subset X=\mathbb{S}^2/{\sim}$ and $V\subset X$ such that $X=U\cup V$, I have initially taken $U=X-\{(0,0,1)\}$ and $V=X-\{(1,0,0)\}$, but I do not know how these spaces look, could someone tell me which spaces are homotopic $U$ and $V$? Thank you.