Spaces determined by countably many locally compact subspaces (Q6581823)

A topological space \(X\) is called an \(\ell c_\omega \)-space if there is an increasing sequence \((K_{n})_{n \in \mathbb{N}}\) of locally compact spaces such that their union equals \(X\), their inclusions \(K_{n} \rightarrow K_{n+1}\) are continuous, and a subset \(U\) of \(X\) is open if and only if \( U \cap K_{n} \) is open in \(K_{n}\) for each \( n \in \mathbb{N} \). In contrast with compactly-generated spaces, this property is preserved by forming finite products. To study posets equipped with the Scott topology, the authors introduce the notion of an \(\ell c_\omega \)-poset and other related concepts. It is proved that for \(\ell c_\omega \)-posets \(P\) and \(Q\), the product \( P \times Q \) is an \(\ell c_\omega \)-poset, the Scott space \(\Sigma P\) is an \(\ell c_\omega \)-space, and \( \Sigma (P \times Q) = \Sigma P \times \Sigma Q \). As an application, conditions are investigated under which the Scott space of a dcpo \(P\) is sober. A particular such one is that \(P\) is an \(\ell c_\omega \)-poset and \( \Sigma P \) is a strong \(d\)-space, which is a strengthened version of a \(d\)-space (monotone convergence space).