Dependent subspaces in C _pC _p(X) and hereditary cardinal invariants

In this paper, for a given arbitrary subset B с 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 с X satisfying the conditions hl(A) < hl(B), hd(A) < hd(B), and s(A) < 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 ^ 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) с 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) < т, hd(B) < т, s(B) <т 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° _pC _p (X) (see Corollary 2.6).

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