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).
@article{VTGU_2015_1_a0,
author = {V. R. Lazarev},
title = {Dependent subspaces in $C_pC_p(X)$ and hereditary cardinal invariants},
journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
pages = {5--11},
year = {2015},
number = {1},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VTGU_2015_1_a0/}
}
TY - JOUR
AU - V. R. Lazarev
TI - Dependent subspaces in $C_pC_p(X)$ and hereditary cardinal invariants
JO - Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
PY - 2015
SP - 5
EP - 11
IS - 1
UR - http://geodesic.mathdoc.fr/item/VTGU_2015_1_a0/
LA - ru
ID - VTGU_2015_1_a0
ER -
%0 Journal Article
%A V. R. Lazarev
%T Dependent subspaces in $C_pC_p(X)$ and hereditary cardinal invariants
%J Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
%D 2015
%P 5-11
%N 1
%U http://geodesic.mathdoc.fr/item/VTGU_2015_1_a0/
%G ru
%F VTGU_2015_1_a0
V. R. Lazarev. Dependent subspaces in $C_pC_p(X)$ and hereditary cardinal invariants. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 1 (2015), pp. 5-11. http://geodesic.mathdoc.fr/item/VTGU_2015_1_a0/
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[2] Arkhangel'skiy A. V., Topologicheskie prostranstva funktsiy, Moskow St. Univ. Publ., M., 1989 (in Russian)
[3] Okunev O., “Homeomorphisms of function spaces and hereditary cardinal invariants”, Topol. and it's Appl., 80 (1997), 177–188 | DOI | Zbl
[4] Lazarev V. R., “O modifikatsii ponyatiya funktsionala s konechnym nositelem”, Vestnik Tomskogo gosudarstvennogo universiteta, 2007, no. 298, 119–120 (in Russian) | Zbl
[5] Lazarev V. R., “O prostranstve funktsionalov s konechnym nositelem”, Vestn. TGU. Byull. operativnoy nauchnoy informatsii «Aktual'nye problemy algebry i analiza», 2005, no. 54, 80–87 (in Russian)
[6] Tkachuk V. V., “Some non-multiplicative properties are l-invariant”, Comment. Math. Univ. Carolin., 38:1 (1997), 169–175 | Zbl
