Consider the equivalence relation on $S^2$ define by $x\sim -x$ if $x\in S^{1}$ (we are supposed to see $S^1\hookrightarrow S^2$ as the equator) and $x\sim x$ otherwise. I have some questions concerning the quotient space $S^2/\sim$:
Is $S^2/\sim$ a $CW$-complex?
If so, what is the cell structure on it?
I'm not an expert but I think it is a CW-complex. You will have one $0$-cell, one $1$-cell and two $2$-cells. The function used to glue the two $2$-cells to $S^1$ is the same and it is $z\mapsto z^2$ (if you think $S^1\subseteq\mathbb{C}$).
I see this very similar to the CW-structure on $\mathbb{P}^2(\mathbb{R})$.