Attaching $2$-dimensional cell to $D^2$ gives the space $S^2/(x\sim -x)$

Question

I am studying Algebraic Topology, and right now I am going through cell-attachment, which I have a pretty hard time to grasp. An "example" they give in the book is:

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Example:

Define $X$ to be the space obtained from $S^2$ by identifying antipodal points on the equator, then it is easy to see that $X$ can be obtained by attaching a $2$-dimensional cell to $D^2$.

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I suppose this should be easy, but it isn't for me, so I would be really happy if someone could help me through this example and how to define the map. Since I don't even know where (and how) to start.

Answer 1

Take $D^2$ and let the attaching map be $e:\partial D^2 \to \partial D^2$ be the quotient map, as in $z \mapsto z^2 $ as complex numbers.

Also note that if the attaching map were identity, we would just recover $S^2$.