Quotient compact images of metric spaces, and related matters (Q1612254)

Some characterizations for the quotient compact images of metric spaces are obtained by means of weak bases. In this paper, the following notion of \(\sigma\)-strong networks as a generalization of development in developable spaces is given: Let \(\{{\mathcal C}_n:n \in\mathbb{N}\}\) be a sequence of covers of a space \(X\) such that \({\mathcal C}_{n+1}\) refines \({\mathcal C}_n\) for each \(n\in\mathbb{N}\), then \(\bigcup\{{\mathcal C}_n:n \in\mathbb{N}\}\) is called a \(\sigma\)-strong network for \(X\) if \(\{st(x,{\mathcal C}_n):n \in\mathbb{N}\}\) is a local network at \(x\) in \(X\). New characterizations for quotient compact images of metric spaces by means of \(\sigma\)-strong networks are obtained. The main theorem is that the following are equivalent for a Hausdorff space \(X\): (1) \(X\) is a sequence-covering, quotient compact-image of a metric space; (2) \(X\) has a point-regular weak base; (3) \(X\) is a sequential space with a \(\sigma\)-point-finite strong cs-network. Finally, the authors pose an interesting question: For a sequential space \(X\) with a point-regular \(\text{cs}^*\)-network, characterize \(X\) by means of a nice image of a metric space.