**Definition 1.** A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called *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.

**Definition 2.** A compact Hausdorff space $X$ is called *Parovichenko* (resp. *soft Parovichenko*) if $X$ is homeomorphic to the remainder $c\mathbb N\setminus\mathbb N$ of some (soft) compactification $c\mathbb N$ of $\mathbb N$?

>**Problem 1.** Is each Parovichenko compact space soft Parovichenko?

**Remark.** The Stone-Cech compactification $\beta\mathbb N$ of $\mathbb N$ is soft, but there are [simple examples][1] of compactifications which are not soft. 

The following concrete version of Problem 1 describes an example of a Parovichenko space for which we do not know if it is soft Parovichenko.

>**Problem 2.** Let $X$ be a compact space that can be written as the union $X=A\cup B$ where $A$ is homeomorphic to $\beta\mathbb N\setminus\mathbb N$, $B$ is homeomorphic to the Cantor cube $\{0,1\}^\omega$ and $A\cap B\ne\emptyset$. Is the space $X$ soft Parovichenko?

  [1]: https://mathoverflow.net/questions/309458/is-each-compactification-of-mathbb-n-soft