Let $p,q\in \beta \mathbb{N}\setminus \mathbb{N}$. Must always the spaces $\beta \mathbb{N}\setminus \{p\}$ and $\beta \mathbb{N}\setminus \{q\}$ be homeomorphic? If no, can we for each point $p\in \beta \mathbb{N}\setminus \mathbb{N}$ find $q\in \beta \mathbb{N}\setminus \mathbb{N}$ ($q\neq p$) such that $$\beta \mathbb{N}\setminus \{p\}\approx \beta \mathbb{N}\setminus \{q\}?$$

Or maybe there exists an uncountable family $F\subset \beta \mathbb{N}\setminus \mathbb{N}$ such that the spaces corresponding to points in $F$ are mutually non-homeomorphic?