Does the space $\mathbb{Q} \times \mathcal{C}$ possess a complete sequence of $\sigma$-discrete closed covers?
I am interested in this question, because if answered positively, the Theorem 1 in the paper by Junnila & Künzi could be used to characterize the space $\mathbb{Q} \times \mathcal{C}$ .
But I am not sure whether this has been answered, haven´t found it in any literature.
Definitions
If $S \subseteq P(X)$ for a topological space $X$, then we say that $S$ is $\sigma$-discrete if $S = \bigcup\limits_{n=1}^{\infty} S_{n}$ where each $S_n$ is discrete.
$\mathbb{Q}\times\mathcal{C}$ = Product of rational numbers and the Cantor set
Complete sequence of covers: A sequence $(Fn)_{n \in \mathbb{N}}$ of covers of a topo. space $X$ is said to be complete, if every filter which intersects all $F_n$´s has an accumulation point in $X$.
Filter on X = family of subsets of X, which is closed with respect to supersets and finite intersections and does not contain the empty set
A point $x \in X$ is said to be an accumulation point of a filter $F$ on X, if each neighborhood of $x$ intersects each element of $F$.
It seems to me that the countable, hence $\sigma$-discrete, cover $\bigl\{\{q\}\times\mathcal{C}:q\in\mathbb{Q}\bigr\}$ gives a constant sequence that is complete: if a filter intersects the cover then it has a compact element and hence accumulation points.