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This question is from the Book [An introduction to Manifolds], Loring W. Tu. p.80.

I understand almost all the other steps of constructing charts for real projective space, $\mathbb{R}P^n$. However, I've got some problem to understand the following :

Let $U_0$ be $\{[a^0, \ldots, a^n] \in \mathbb{R}P^n | a^0 \neq 0 \}$, and define $f : \mathbb{R}^n \rightarrow U_0$ by $(a^1, \ldots, a^n) \mapsto [1, a^1, \ldots, a^n]$. Then $f$ is continuous.

Please let me know this. Thank you.

with-forest
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1 Answers1

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Define $g:\mathbb{R}^n \to \mathbb{R}^{n+1}$ by $g(a^1,\dots,a^n)=(1,a^1,\dots,a^n)$. If $\pi:\mathbb{R}^{n+1} \to \mathbb{RP}^n$ is the natural projection, then $f=\pi \circ g$ is the composition of continuous maps.

Bob
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