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Metrizability, generalized metric spaces and g-functions. (English) Zbl 0686.54019

Let (X,\(\tau)\) be a topological space and \({\mathbb{N}}\) denote the natural numbers, a function g:\({\mathbb{N}}\times X\to \tau\) is called a g-function if each g(n,x) is a neighborhood of x (in most cases, it is not necessary that g(n,x) be open). The author provides a rather complete survey of characterizations of generalized metric spaces in terms of g-functions; there are also some new results. Theorem. A space X is metrizable iff X has a g-function which satisfies (1) for each \(x\in X\) and nbhd P of x there is \(n\in {\mathbb{N}}\) such that \(x\not\in (\cup \{g(n,y)| y\in X- P\})^-\) and (2) for each \(Y\subset X\), \(\bar Y\subset \cup \{g^ 2(n,y)| y\in Y\}\), where \(g^ 2(n,y)=\cup \{g(n,z)| z\in g(n,y)\}\). There are other results and several open questions.
Reviewer: C.R.Borges

MSC:

54E35 Metric spaces, metrizability
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