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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
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