Categorical Structure of Closure Operators

Dikranjan, Dikran; Tholen, Walter

doi:10.1007/978-94-015-8400-5

Cited by 149 publications

(182 citation statements)

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“…For the general categorical terminology used see [1] and for that concerning categorical closure operators see [4] and [18]. The lattice-theoretic concepts and results used are taken from [24] and the topological ones from [19].…”

Section: Preliminariesmentioning

confidence: 99%

“…Basic examples of the above introduced category K with a closure operator are certain topological constructs with X = Set where | |: K → Set is the forgetful functor and the (surjections, injections)-factorization structure for morphisms is considered in the base category Set. A number of such examples are given in [4], [5], [11], [12], [18]. Among them, of course, the most natural one is K = Top, i.e., the construct of topological spaces and continuous maps, with c the Kuratowski closure operator.…”

Section: Neighborhoods and Convergence With Respect To A Closure Opermentioning

confidence: 99%

“…Among them, of course, the most natural one is K = Top, i.e., the construct of topological spaces and continuous maps, with c the Kuratowski closure operator. In what follows, if K = Top or a (full) subconstruct of Top is taken as an example of K , we always mean that c is just the Kuratowski closure operator (of course, there are also other closure operators on Top -see [18]). Further examples can be found among concrete categories over topological constructs (with a singleton fibre of the empty set) which always have the (surjections,embeddings)-factorization structure for morphisms.…”

Section: Neighborhoods and Convergence With Respect To A Closure Opermentioning

confidence: 99%

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Neighborhoods and convergence with respect to a closure operator

Šlapal

2011

Mathematica Slovaca

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ABSTRACT. We study neighborhoods with respect to a categorical closure operator. In particular, we discuss separation and compactness obtained from neighborhoods in a natural way and compare them with the usual closure separation and closure compactness. We also introduce a concept of convergence based on using centered systems of subobjects, which naturally generalizes the classical filter convergence in topological spaces. We investigate behavior of the convergence introduced and show, among others, that it relates to the separation and compactness in natural ways.

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Section: Preliminariesmentioning

confidence: 99%

Section: Neighborhoods and Convergence With Respect To A Closure Opermentioning

confidence: 99%

Section: Neighborhoods and Convergence With Respect To A Closure Opermentioning

confidence: 99%

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Neighborhoods and convergence with respect to a closure operator

Šlapal

2011

Mathematica Slovaca

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“…A topological space is an Alexandroff space if the intersection of any family of open sets is open (resp., the union of any family of closed sets is closed) [3].…”

Section: Proof Suppose That δ Separates S a Primary Sense The Relatmentioning

confidence: 99%

Mathematical morphology and poset geometry

Bretto¹,

Marzi

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. The aim of this paper is to characterize morphological convex geometries (resp., antimatroids). We define these two structures by using closure operators, and kernel operators. We show that these convex geometries are equivalent to poset geometries.2000 Mathematics Subject Classification. 37F20, 06A07. Introduction.Convex geometries are particular closure operators. These objects link the theory of convex sets to combinatorial theory. More precisely poset geometries are basic structures, because any convex geometry can be generated from them [6].Mathematical morphology, introduced by Serra [7] is a very important tool in image processing and pattern recognition. The framework of mathematical morphology consists in erosions and dilations (resp., Galois functions) which result in morphological closures and kernels. These particular operators are well adapted to algorithmic computation [5].Theorem 4.2 proves that poset geometries and morphological geometries defined below are equivalent. The convexity is very important in mathematical morphology [5]. So this article tries to demonstrate the relation between combinatorial convexity, mathematical morphology, and image processing.

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“…The category of filter convergence spaces and continuous maps is denoted by FCO (see [15] p.45 or [30] p.354). A filter convergence space (A, L) is said to be a local filter convergence space (in [29], it is called a convergence space…”

Section: Preliminariesmentioning

confidence: 99%