Some properties of \(\sigma\)-products (Q1801716)

Definition 1: Let \(\{X_t : t \in T\}\) be a family of topological spaces and \(p = (p_t) \in \Pi X_t\). The \(\sigma\)-product of \(X_t\), \(t \in T\), having base point \(p\), is defined by \(X = \{x \in \Pi X_t : |\{t \in T : x_t \neq p_t\} |< \aleph_0\}\). Notation, \(X = \sigma \{X_t : t \in T\}\). Definition 2: A space \(X\) is almost \(\theta\)-expandable if for every locally finite family \(\{F_\alpha : \alpha \in A\}\) of closed subsets of \(X\) there exists a sequence \(\langle {\mathcal G}_n = \{G_{n \alpha} : \alpha \in A\} \rangle_{n < w}\) of collections of open subsets of \(X\) satisfying the following: (1) \(F_\alpha \subset G_{n \alpha}\) for each \(\alpha \in A\) and \(n < w\). (2) For each \(x \in X\) there is some \(n < w\) such that \({\mathcal G}_n\) is point finite at \(x\). In this paper the author shows the following: Theorem 1: Let \(X = \sigma \{X_t : t \in T\}\) be a \(\sigma\)-product space. If every finite subproduct of \(X\) is almost \(\theta\)-expandable, then \(X\) is almost \(\theta\)-expandable. Theorem 2: For any space \(X\) the following are equivalent: (1) \(X\) is almost \(\theta\)-expandable. (2) Every \(A\)-cover of \(X\) has an open pointwise \(W\)-refining sequence. (3) Every \(A\)-cover of \(X\) has a semi-open point-star \(F\)-refining sequence. (4) Every directed \(A\)-cover of \(X\) has a semi-open point-star refining sequence. (5) Every directed \(A\)-cover of \(X\) has a \(\sigma\)-closure-preserving closed refinement. Corollary: The continuous closed image of an almost \(\theta\)-expandable space is almost \(\theta\)-expandable. Similar results are shown for \([\theta,k]\)-compactness.