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This is a ``1-soft'' modification of this problem. We start with the necessary definitions.

Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called 1-soft if for any subset $A\subset\mathbb N$ with $\overline{A}\cap\overline{\mathbb N\setminus A}\ne\emptyset$ there exists a homeomorphism $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\setminus\mathbb N$ and the set $\{x\in A:h(x)\notin A\}$ is infinite.

Definition 2. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called 2-soft if for any disjoint sets $A,B\subset\mathbb N$ with $\bar A\cap\bar B\ne\emptyset$ there exists a homeomorphism $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\setminus\mathbb N$ and the set $\{x\in A:h(x)\in B\}$ is infinite.

Before the formulation of a question, let us recall some known results.

Theorem (Parovichenko). Each compact Hausdorff space $K$ of weight $\le\omega_1$ is homeomorphic to the remainder $c\mathbb N\setminus \mathbb N$ of some compactification $c\mathbb N$ of $\mathbb N$.

Theorem (Hart). Under CH, each compact Hausdorff space $K$ of weight $\le\omega_1$ is homeomorphic to the remainder of a 2-soft compactification of $\mathbb N$.

Example (Dow). Under (NT) the compact space $K=\omega_1+1+\omega_1^*$ is not homeomorphic to the remainder of a 2-soft compactification of $\mathbb N$.

Question 1. Is the compact space $K=\omega_1+1+\omega_1^*$ homeomorphic to the remainder of a 1-soft compactification of $\mathbb N$?

Question 2. Is each compact Hausdorff space of weight $\le\omega_1$ homeomorphic to the remainder of a 1-soft compactification of $\mathbb N$?


Added in Edit. The answer to Question 1 is affirmative. So, only Question 2 remains open.

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  • $\begingroup$ I guess the $B$ in the definition of $1$-soft should be $A$? $\endgroup$ – KP Hart Jan 30 at 16:32
  • $\begingroup$ Yes, exactly! Thank you (I am glad that you noticed this question). $\endgroup$ – Taras Banakh Jan 30 at 17:17
  • $\begingroup$ @KPHart I have just asked another set-theoretic question (mathoverflow.net/q/351555/61536) related to Question 1. $\endgroup$ – Taras Banakh Jan 30 at 17:52
  • $\begingroup$ Could you include the construction for 1? $\endgroup$ – KP Hart Feb 27 at 12:57
  • $\begingroup$ @KPHart I added a link to the paper in arXiv. $\endgroup$ – Taras Banakh Feb 27 at 15:34

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