Are the remaiders $\beta \mathbb{N} - \mathbb{N}$ and $\beta K_0 - K_0$ homeomorphic?

Question

Are the remainders $\beta \mathbb{N} - \mathbb{N}$ and $\beta K_0 - K_0$ homeomorphic (where $K_0$ means the Cantor set without a point and $\beta$ means the Stone-Cech compactification)?

Answer 1

Under CH (the continuum hypothesis) the answer is yes: theorem 1.2.6.in the introduction to $\beta \omega$ has the following corollary to Paravicenko's theorem under CH:

(CH) If $X$ and $Y$ are zero-dimensional, locally compact, non-compact, $\sigma$-compact spaces of weight at most $\mathfrak{c}$, then $\beta X - X$ and $\beta Y - Y$ are homeomorphic.

Note that $\mathbb{N}$ and $K_0$ are such spaces.

Maybe ideas like these work: we can write $K_0 = \cup_n C_n$ all clopen compact and pairwise disjoint, and map $C_n$ to $n$ and extend to $\beta \mathbb{N}$. This maps $\beta K_0 - K_0$ onto $\beta \mathbb{N} - \mathbb{N}$ (by perfectness, e.g.). It this map were 1-1, we'd have an absolute homeomorphism, not dependent on CH.