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Characterizations of Urysohn-closed spaces. (English) Zbl 0785.54025

In [Proc. Am. Math. Soc. 55, 435-439 (1976; Zbl 0354.54010)] L. L. Herrington obtained characterizations of Urysohn-closed spaces in terms of arbitrary filterbases and a type of convergence for filterbases called \(u\)-convergence. J. E. Joseph [ibid. 68, 235-242 (1978; Zbl 0407.54012)] utilized these characterizations to obtain characterizations of of these spaces in terms of graphs of functions and in terms of projections into the space. In [Proc. Natl. Acad. Sci. India, Sect. A 54, 431-437 (1984; Zbl 0599.54028)] the second author and G. Lo Faro obtained characterizations of Urysohn-closed spaces in terms of a type of cover as Urysohn weakly compact spaces [Riv. Mat. Univ. Parma, IV. Ser. 7, 383-395 (1981; Zbl 0506.54018)] and a type of convergence for open filterbases called \(\gamma\)-convergence and in terms of adherence of a type of open filterbases called almost-regular.
In the present article, our primary goal is to give a characterization of Urysohn-closed spaces. Initially, we introduce \(\nu\)-continuous functions and give some characterizations of such functions and weakly-compact spaces using filterbases and nets. These characterizations are obtained mainly through the introduction of a type of convergence, adherence for filterbases and nets called \(\nu\)-convergence (\(\nu\)-adherence). In the last section we define the functions with \(\nu\)-closed graph and study the relationship between such functions and functions with stronger- closed graph [L. L. Herrington and P. E. Long, Proc. Am. Math. Soc. 48, 469-475 (1975; Zbl 0306.54032)]. Hence using the technique of [S. Kasahara, Proc. Japan Acad. 49, 523-524 (1973; Zbl 0273.54014)] we obtain a characterization of Urysohn-closed spaces. Let \(S\) denote a class of topological spaces containing the class of Hausdorff completely normal and fully normal spaces. This characterization is then stated as follows: A Urysohn space \(Y\) is Urysohn-closed if and only if for every space \(X\) in the class \(S\), each \(f:X\to Y\) with a \(\nu\)-closed graph is \(\nu\)-continuous.

MSC:

54D25 “\(P\)-minimal” and “\(P\)-closed” spaces
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
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