# A 1-soft'' improvement of the Parovichenko theorem

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.

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