Theory of \(k\)-networks. II (Q2710540)

Let \(X\) be a space, and \({\mathcal P}\) a cover of \(X\). \({\mathcal P}\) is said to be a \(k\)-network for \(X\) if \(K\subset U\) with \(K\) compact and \(U\) open, then \(K\subset \bigcup{\mathcal P}'\subset U\) for some finite subset \({\mathcal P}'\) of \({ \mathcal P}\). \(k\)-networks, as a generalization of bases of a space, have become an important tool for the study of general topology. The author's paper [ibid. 12, No. 2, 139-164 (1994; Zbl 0833.54015)] gave a nice survey of the theory of \(k\)-networks, including some relationships, mapping theorems and product theorems among generalized metric spaces defined by \(k\)-networks, and questions. In this paper, the author gives a survey of \(k\)-networks among generalized metric spaces, which contains a very detailed and well-organized exposition of recent new results, including spaces with a star-countable \(k\)-network, \(k\)-spaces of product spaces with certain \(k\)-networks, and \(k\)-networks in a CW-complex and a topological group, etc.NEWLINENEWLINENEWLINEThe reviewer highly recommends this paper to anyone seriously interested in \(k\)-networks and questions.