Spaces with compact-countable weak-bases

In this paper, we establish the relationships between spaces with a compact-countable weak-base and spaces with various compact-countable networks, and give two mapping theorems on spaces with compact-countable weakbases. Weak-bases and g-first countable spaces were introduced by A.V.Arhangel’skii [1]. Spaces with a point-countable weak-base were discussed in [5,6], and spaces with a locally countable weak-base were discussed in [7,8,9]. In this paper, we shall investigate spaces with a compact-countable weak-base, establish the relationships between spaces with a compact-countable weak-base and spaces with various compact-countable networks, and give two mapping theorems on spaces with compact-countable weak-bases. We assume that spaces are regular and T1, and mapping are continuous and onto. Definition 1. Let P be a family of subsets of a space X, put P = {P ′ ⊂ P : |P ′| < ω}. (1) P is compact-countable in X if for each compact subset K of X, only countably many members of P intersect K. (2) P is a k-network for X if for each compact subset K of X and its open neighborhood V , there exists P ′ ∈ P such that K ⊂ ∪P ′ ⊂ V . (3) P is a cs-network for X if for each x ∈ X, its open neighborhood V and a sequence {xn} converging to x, there exists P ∈ P such that {xn : n > m} ∪ {x} ⊂ P ⊂ V for some m ∈ N . Definition 2. For a space X and x ∈ P ⊂ X, P is a sequential neighborhood of x in X if, whenever {xn} is a sequence converging to x in X, then xn ∈ P for all but finitely many n ∈ N . P is a sequential open set of X if for each x ∈ P , P is a sequential neighborhood of x in X. A space X is a sequential space if each sequential open set of X is open in X. 2000 Mathematics Subject Classification. 54D50; 54D99; 54E40.