Finite compactifications of $ω^* \setminus \{x\}$

We prove that under [CH], finite compactifications of $ω^* \setminus \{x\}$ are homeomorphic to $ω^*$. Moreover, in each case, the remainder consists almost exclusively of $P$-points, apart from possibly one point. Similar results are obtained for other, related classes of spaces, amongst them $S_κ$, the $κ$-Parovičenko space of weight $κ$. Also, some parallels are drawn to the Cantor set and the Double Arrow space.

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