**Question**. Let $\kappa>\mathfrak c$ be a cardinal endowed with the discrete topology and $f:\kappa^\omega\to[0,1]^\omega$ be a continuous map. Is there a point $y\in[0,1]^\omega$ whose preimage $f^{-1}(y)$ is not scattered?

Let us recall that a topological space $X$ is *scattered* if each non-empty subspace of $X$ contains an isolated point.

An affirmative answer to this question would give an affirmative answer to this problem.

**Remark.** If $\kappa<\kappa^\omega$, then the answer to the above question is affirmative. Indeed, assuming that every preimage $f^{-1}(y)$, $y\in[0,1]^\omega$, is scattered, we conclude that $|f^{-1}(y)|\le w(f^{-1}(y))\le w(\kappa^\omega)=\kappa$ and hence $|\kappa^\omega|\le\sum_{y\in[0,1]^\omega}|f^{-1}(y)|\le\mathfrak c\cdot \kappa=\kappa<|\kappa^\omega|$, which is a desired contradiction. So, actually, the question concerns cardinals $\kappa=\kappa^\omega>\mathfrak c$. The smallest such cardinal is $\mathfrak c^+$, the successor of the continuum.