Q1: What is the definition of the $n$-dimensional sphere?

A1: $S^n = { x \in \mathbb{R}^{n+1} \mid | x | \leq 1 }$

A2: $S^n = { x \in \mathbb{R}^n \mid | x | \geq 1 }$

A3: $S^n = { x \in \mathbb{R}^{n+1} \mid | x | = 1 }$

A4: $S^n = { x \in \mathbb{R}^{n} \mid | x | = 1 }$

A5: $S^n = { x \in \mathbb{R}^{n+1} \mid | x | \geq 1 }$

Q2: 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 $.

Q3: Is the projective space $\mathbf{P}^n(\mathbb{R})$ together with the quotient topology a Hausdorff space?

A1: Yes!

A2: No!

A3: Only for $n=1$.