Q1: What is the definition of the projective space $\mathbf{P}^n(\mathbb{R})$?

A1: $S^n! /! ! \sim ~~$ for $x\sim y$ if $x=y$.

A2: $S^n! /! ! \sim ~~$ for $x\sim y$ if $x=y$ or $x = -y$.

A3: $S^n! /! !\sim ~~$ for $x\sim y$ if $x=y$ and $x = -y$.

A4: $S^n ! /! ! \sim ~~$ for $x\sim y$ if $x=y-1 $.

Q2: Is the projective space $\mathbf{P}^n(\mathbb{R})$ together with the quotient topology a manifold?

A1: Yes!

A2: No!

A3: Only for $n=1$.

Q3: Let $\mathbf{P}^n(\mathbb{R})$ be the projective space. What is the dimension of the manifold?

A1: $n-1$

A2: $n$

A3: $n+1$