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Regular \(\mathcal L \)-spaces. (English) Zbl 0797.54005
Recently the authors introduced the notion of \(\mathcal L\)-regularity of sequential convergence spaces. In this paper they continue to investigate \(\mathcal L\)-regularity, in particular in the class of \({\mathcal L}^{\text{f}}\)-spaces. (An \(\mathcal L\)-space \((X,L)\) is said to be an \({\mathcal L}^{\text{f}}\)-space if there exists a filter convergence structure \(q\) on \(X\) such that \(L\) is associated with \(q\), i.e. \(L = {\mathcal L}(q)\).) For example they prove that an \({\mathcal L}^{\text{f}}\)- space \(X\) is \(\mathcal L\)-regular iff its \(\gamma\)-modification \(\gamma X\) (in the sense of Beattie and Butzmann) is a regular convergence space. A characterization of \(\mathcal L\)-regular \({\mathcal L}^*\)-spaces is given. Regular modifications of \({\mathcal L}^{\text{f}}\)-spaces and \({\mathcal L}^*\)-spaces are defined and studied and some categorical aspects of such modifications are also considered. Some characterizations of \({\mathcal L}^{\text{f}}\)-spaces having regular modifications with unique limits are given. The authors also consider the (topological) category \(\mathcal L\) of \(\mathcal L\)-spaces and \(\mathcal L\)-continuous mappings and some of its full subcategories.

MSC:
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54B30 Categorical methods in general topology
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References:
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