Openness with respect to a closure operator
The abstract notion of a closure operator \(c\) introduced by \textit{E. Giuli} and \textit{D. Dikranyan} [Topology Appl. 27, 129-143 (1987; Zbl 0634.54008)] permits to define easily the notion of \(c\)-closed map (loc. cit.). The authors study \(c\)-open maps [introduced by \textit{W. Tholen} and \textit{D. Dikranyan}, ``Categorical structure of closure operators with applications to topology, algebra and discrete mathematics'', Mathematics and its Applications, vol. 346, Kluwer Academic Publishers, Dordrecht-Boston-London (1995; Zbl 0853.18002)], as well as \(c\)-initial maps [introduced by the reviewer, \textit{D. Dikronyan}, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 29, 105-160 (1992; Zbl 0782.54009)] and \(c\)-final maps [introduced by \textit{M. M. Clementino}, Appl. Categ. Struct. 4, No.~1, 15-29 (1996; Zbl 0883.54013)]. The paper studies mainly the pullback behaviour of openness/closedness and initiality/finality with respect to an arbitrary closure operator \(c\). Another important contribution of this paper is the isolation of the notion of \(c\)-open subobject within the stronger property of \(c\)-open morphism. A lot of examples of topological and algebraic nature are given. This paper should be compared with another recent paper of \textit{M. M. Clementino, E. Giuli} and \textit{W. Tholen} [Appl. Categ. Struct. 9, No. 2, 139-151 (2001; Zbl 0978.18003)] where they show that the analogous problem with respect to quotient maps has not a satisfying solution.
scientific article; zbMATH DE number 1563363