Abstract
It is well known that topological spaces are axiomatically characterized by the topological closure operator satisfying the Kuratowski Closure Axioms. Equivalently, they can be axiomatized by other set operators encoding primitive semantics of topology, such as interior operator, exterior operator, boundary operator, or derived-set operator (or dually, co-derived-set operator). It is also known that a topological closure operator (and dually, a topological interior operator) can be weakened into generalized closure (interior) systems. What about boundary operator, exterior operator, and derived-set (and co-derived-set) operator in the weakened systems? Our paper completely answers this question by showing that the above six set operators can all be weakened (from their topological counterparts) in an appropriate way such that their inter-relationships remain essentially the same as in topological systems. Moreover, we show that the semantics of an interior point, an exterior point, a boundary point, an accumulation point, a co-accumulation point, an isolated point, a repelling point, etc. with respect to a given set, can be extended to an arbitrary subset system simply by treating the subset system as a base of a generalized interior system (and hence its dual, a generalized closure system). This allows us to extend topological semantics, namely the characterization of points with respect to an arbitrary set, in terms of both its spatial relations (interior, exterior, or boundary) and its dynamic convergence of any sequence (accumulation, co-accumulation, and isolation), to much weakened systems and hence with wider applicability. Examples from the theory of matroid and of Knowledge/Learning Spaces are used as an illustration.
Highlights
Let us recall the notions of topology and topological space [1,2,3]
We remark that the above characterizations of set operators in terms of points were defined in any subset system ( X, S), where X is a set and S is a collection of its subsets
The usual semantics of closure, interior, exterior, boundary, derived set, and co-derived set can be used in subset systems more widely
Summary
Let us recall the notions of topology and topological space [1,2,3]. A topology on a set X is a collection T of subsets of X, including the empty set ∅ and X itself, in which T is closed under arbitrary union and finite intersection; ( X, T ) is called a topological space. We can take set-complement of each of these closed sets to obtain another collection (i.e., another system of sets), which properly form an (open set) topology In this sense, we can say that an operator satisfying the Kuratowski Closure Axioms [CO1]–[CO4] defines a topological space ( X, T ). Topology [12,13], where the categorical closure operator studied are often not even required to satisfy the (categorical analogue of the) axiom [CO3] Given this theoretical backdrop, we can immediately ask whether there exist meaningful generalizations of the other five operators (interior, exterior, boundary, derived set, co-derived set) of a Topological System to any Closure System.
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