As I have to demonstrate that this space is a Hausdorff space, obviously I hope to find a proof without the use of this fact
2026-03-30 04:56:41.1774846601
How can I show that the quotient function to the real projective space is closed?
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I'm assuming by quotient function, you mean the one from $S^n \rightarrow \mathbb{R}P^n$? Regardless, it makes no difference to the outcome. The way that the quotient function/topology is defined gives you the result.
Edit: As David pointed out, quotient maps are not always closed. However, due to the specifics of how $\mathbb{R}P^n$ is defined using the quotient, it actually is in this case. But then this is just restating the question. My comments about the CW complex still stands.
If you are trying to show that $\mathbb{R}P^n$ is Hausdorff, you can construct a CW complex structure for it since any CW complex is Hausdorff. This may be simply moving the goal posts in your case though if you haven't seen this result yet. When I have time, I can find a link to another post on this site showing CW complexes are Hausdorff, there's bound to be at least 1.