I found this question in a book (Topology II: homotopy and homology: classical manifolds)
Show that the quotient space $X = S^2 \times S^2 / [(x_1,x_2) \sim (Rx_1,Rx_2)]$ where R is the reflection in the equatorial plane, is homeomorphic to $S^4$.
I am still in the process of learning topology and I really don't think I can prove this result (or even understand a proof if someone were generous enough to provide me with one). I apologize in advance if this is trivial, but it will be of great help if someone could give a homeomorphism. I would really like to use the mapping of the space $X$ to $S^4$ in my work. If you are aware of the proof and would provide it in your answer, I will definitely make an effort to understand it.
Thank you for your time.