Codenseness and Openness with Respect to an Interior Operator
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Codenseness and Openness with Respect to an Interior Operator Fikreyohans Solomon Assfaw1
· David Holgate1
Received: 13 September 2019 / Accepted: 9 October 2020 © Springer Nature B.V. 2020
Abstract Working in an arbitrary category endowed with a fixed (E , M)-factorization system such that M is a fixed class of monomorphisms, we first define and study a concept of codense morphisms with respect to a given categorical interior operator i. Some basic properties of these morphisms are discussed. In particular, it is shown that i-codenseness is preserved under both images and dual images under morphisms in M and E , respectively. We then introduce and investigate a notion of quasi-open morphisms with respect to i. Notably, we obtain a characterization of quasi i-open morphisms in terms of i-codense subobjects. Furthermore, we prove that these morphisms are a generalization of the i-open morphisms that are introduced by Castellini. We show that every morphism which is both i-codense and quasi i-open is actually i-open. Examples in topology and algebra are also provided. Keywords Interior operator · Codenseness · Openness · Quasi-openness Mathematics Subject Classification 06A15 · 18A20 · 54B30
1 Introduction A categorical closure operator on an arbitrary category is a family of functions (on suitably defined subobject lattices) which are expansive, order preserving and compatible with taking images or equivalently, preimages, in the same way as the usual topological closure is compatible with continuous maps. The formal theory of categorical closure operators was introduced by Dikranjan and Giuli [11] and then developed by these authors and Tholen [12]. The theory was largely inspired by Salbany’s paper [23], where regular closure operators on the category of topological spaces and continuous maps were introduced. These operators
Communicated by Maria Manuel Clementino.
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Fikreyohans Solomon Assfaw [email protected] David Holgate [email protected]
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Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville 7535, South Africa
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F. S. Assfaw, D. Holgate
have played a vital role in the development of Categorical Topology by introducing topological concepts, such as connectedness, separatedness, compactness, denseness and closedness, in an arbitrary category and they provide a unified approach to various mathematical notions (see [4,13]). Motivated by the theory of categorical closure operators, the categorical notion of interior operators was introduced by [24]. These operators have received more recent attention and a few papers are published on the subject; see [3,5–9,17,20,22]. In general topology, interior and closure characterize each other via set-theoretic complement. More generally, closure and interior operators characterize each other in a category equipped with a categorical transformation operator (see [24]). As a consequence, most of the theory of interior operators can be derived from that of closure operators and vice versa. Nevertheless, the two opera
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