Dependent subspaces in $C_pC_p(X)$ and hereditary cardinal invariants

In this paper, for a given arbitrary subset $B\subset C_pC_p(X)$ consisting of finite support functionals (see Definition 1.1), we prove its continuous factorizability (see Definition 0.3) through some subset $A\subset X$ satisfying the conditions $hl(A)\leqslant hl(B)$, $hd(A)\leqslant hd(B)$, and $s(A)\leqslant s(B)$.
Finite support functionals have some essential properties of linear continuous functionals. In particular, the set $B$ above may be “ranked” by subsets $B_n$ according to the number n of points in the supports of functionals. In addition, the support mapping $s_n: B_n\to E_n(X)$ is continuous (see Lemma 1.6). It permit us to formulate conditions on a topological property that are sufficient for the union $X(B)\subset X$ of the supports of the functionals from $B$ to have this topological property together with $B$ (see Theorem 2.3). Since $B$ admits continuous factorization through $X(B)$ (see Lemma 1.8) and inequalities $hl(B)\leqslant \tau$, $hd(B)\leqslant \tau$, $s(B)\leqslant \tau$ keep true under any operations from the formulation of Theorem 2.3 (see Corollary 2.4), we get a partially positive answer to the Problem 3.3 and Problem 3.4 from [3].
In addition, we extend Corollary 2.4 to all open and all canonical closed subsets of the space $C^0_pC_p(X)$ (see Corollary 2.6).