Show that this topology on the sphere is Hausdorff

Question

Consider the space $S^{n-1}$ in $\mathbb{R}^n$. Put $N= (0,0,\dots, 0,1), S = (0,0, \dots, 0,-1)$ and consider the stereographic projections

$$\phi_N: U_N:= S^{n-1}\setminus \{N\} \to \mathbb{R}^{n-1}$$
$$\phi_S: U_S:= S^{n-1}\setminus \{S\} \to \mathbb{R}^{n-1}$$

Consider the atlas $\mathcal{A} = \{(U_N, \phi_N), (U_S, \phi_S)\}$.

Equip the space $S^{n-1}$ with the natural topology

$$\mathcal{T}:= \{V \subseteq S^{n-1}\mid \phi_N(U_N \cap V), \phi_S(U_S\cap V) \mathrm{\ are \ open \ sets \ in \ \mathbb{R}^{n-1}}\}$$

Is the space $\mathcal{T}$ Hausdorff?

My attempt:

I showed that every two points can be separated by disjoint open sets, except the points $N$ and $S$. How would I show that these poins can be separated by disjoint open sets? My guess is that we have to select two specific open sets.

Answer 1

Take the northern hemisphere $\mathcal{N}=\{(x_1,\ldots,x_n)\in S^{n-1}\,|\,x_n>0\}$ and the southern hemisphere $\mathcal{S}=\{(x_1,\ldots,x_n)\in S^{n-1}\,|\,x_n<0\}$. Then:

• $N\in\mathcal N$;
• $S\in\mathcal S$;
• $\mathcal N$ and $\mathcal S$ are open;
• $\mathcal{N}\cap\mathcal{S}=\emptyset$.