Electronic Version 16 (1991), 253-256

Zi Qiu Yun

All spaces are assumed to be regular and T₁ in this paper. A family F of subsets of a space X is called a k-network if for any compact set K and and open set U which contains K, there exists a finite subfamily F^' such that K 'subset' \cup F^' 'subset or equal' U. A space X with a \sigma-locally finite k-network is called an \aleph-space ([7]). A family F of subsets of X is called a cs-network if for any sequence { x_n : n \in N } which converges to a point x \in U, where U is open in X, there exists an F \in F, such that, F 'subset or equal' U and {x_n : n \in N } is eventually in F ([2]). In [1] it was proved that a space X is an \aleph-space if and only if X has a \sigma-locally finite cs-network, and it is well known that the condition of having a \sigma-hereditarily closure-preserving k-network is strictly weaker than that of being an \aleph-space. But, what about the condition of having a \sigma-hereditarily closure-preserving cs-network? Is it equivalent to the former condition or to the latter one? In this paper, we prove that having a \sigma-hereditarily closure-preserving cs-network is a characterization of \aleph-spaces. As an application of this result, we show that \aleph-spaces are preserved under open and closed mappings, which answers a question raised in [5] (See [5] Question 4.4).

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topology proceedings Electronic Version