Closure operators I

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“…In order to describe the epimorphisms and the extremal monomorphisms in Cl 0 we need the regular closure operator determined by Cl 0 as introduced in [8,9]. Given a closure space X and a subset M ⊂ X one defines the regular closure of M in X as follows.…”

T 2 -objects in topological categories

“…(4). For an arbitrary topological category, it is not known in general whether the closure used in 2.1 is a closure operator in the sense of Dikranjan and Giuli [17] or not. However, it is shown, in [10], that the notions of closedness and strong closedness that are defined in 2.1 form appropriate closure operators in the sense of Dikranjan and Giuli [17] in case the category is one of the categories FCO and LFCO.…”

“…Categorical notions of compactness and Hausdorffness with respect to a factorization structure were defined in the case of a general category by Manes [25] and Herrlich, Salicrup and Strecker [22]. A categorical study of these notions with respect to an appropriate notion of "closedness" based on closure operators (in the sense of [17]) was done in [18] (for the categories of various types of filter convergence spaces) and [14] (for abstract categories). Baran in [2] and [4] introduced the notion of "closedness" and "strong closedness" in set-based topological categories and used these notions in [7] to generalize each of the notions of compactness and Hausdorffness to arbitrary set-based topological categories.…”

Neighborhoods and convergence with respect to a closure operator

“…It is more general than the classical closure operator introduced in [15], which is obtained when K = X and | | is the identity functor. Using the above concept of a closure operator, we substantially reduce the restriction given by the assumption that f…”