Quotient compact images of metric spaces, and related matters. (English) Zbl 0994.54015

Ikeda, Y.; Liu, C.; Tanaka, Y.

Topology Appl. 122, No. 1-2, 237-252 (2002).

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.

Reviewer: Shou Lin (Fujian)

MSC:

54C10	Special maps on topological spaces (open, closed, perfect, etc.)
54E40	Special maps on metric spaces
54D55	Sequential spaces

Keywords:

quotient map; compact map; sequence-covering map; \(\sigma\)-strong network; weak base; sequential space

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