# Projective spaces - Disjoint union

I was trying to prove that $\mathbb{RP^{3}}$ is homeomorphic to que quotient space of the disjoint union $\mathbb{RP^{2}}$$\bigsqcup$$B$ by the "gluing" $f$: $\partial$$B \rightarrow \mathbb{RP^{2}}, where B is the closed unit ball in \mathbb{R^{3}}. My idea is to construct a quotient map from \mathbb{S^{3}} to the quotient space of \mathbb{RP^{2}}$$\bigsqcup$$B$ which makes the same identifications that the quotient map which constructs $\mathbb{RP^{3}}$, and conclude by a widely-known theorem that these two spaces are, in fact, homeomorphic. I was trying to do it by composition of some continuous functions, using the closed map lemma to help me, but I couldn't go any further. Do you have any hint or suggestion?

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Do you know CW-complexes? –  Chris Gerig Oct 4 '12 at 5:29
No, I haven't studied it yet... –  Br09 Oct 4 '12 at 16:02