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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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