Topic

# Separated sets

About: Separated sets is a research topic. Over the lifetime, 628 publications have been published within this topic receiving 11364 citations.

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TL;DR: In this article, a semi-Open Sets and Semi-Continuity in Topological Spaces (SOCS) model is proposed, which is based on the semi-continuity in topological spaces.

Abstract: (1963). Semi-Open Sets and Semi-Continuity in Topological Spaces. The American Mathematical Monthly: Vol. 70, No. 1, pp. 36-41.

1,630 citations

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TL;DR: A novel theory of topological spatial relations between sets is developed in which the relations are defined in terms of the intersections of the boundaries and interiors of two sets, and it is shown that these relations correspond to some of the standard set theoretical andTopological spatial Relations between sets such as equality, disjointness and containment in the interior.

Abstract: Practical needs in geographic information systems (GIS) have led to the investigation of formal and sound methods of describing spatial relations. After an introduction to the basic ideas and notions of topology, a novel theory of topological spatial relations between sets is developed in which the relations are defined in terms of the intersections of the boundaries and interiors of two sets. By considering empty and non-empty as the values of the intersections, a total of sixteen topological spatial relations is described, each of which can be realized in R 2. This set is reduced to nine relations if the sets are restricted to spatial regions, a fairly broad class of subsets of a connected topological space with an application to GIS. It is shown that these relations correspond to some of the standard set theoretical and topological spatial relations between sets such as equality, disjointness and containment in the interior.

1,541 citations

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TL;DR: It is shown that a soft topological space gives a parametrized family of topological spaces and it is established that the converse does not hold.

Abstract: In the present paper we introduce soft topological spaces which are defined over an initial universe with a fixed set of parameters. The notions of soft open sets, soft closed sets, soft closure, soft interior points, soft neighborhood of a point and soft separation axioms are introduced and their basic properties are investigated. It is shown that a soft topological space gives a parametrized family of topological spaces. Furthermore, with the help of an example it is established that the converse does not hold. The soft subspaces of a soft topological space are defined and inherent concepts as well as the characterization of soft open and soft closed sets in soft subspaces are investigated. Finally, soft T"i-spaces and notions of soft normal and soft regular spaces are discussed in detail. A sufficient condition for a soft topological space to be a soft T"1-space is also presented.

832 citations

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30 Oct 1996

TL;DR: In this paper, the real numbers linearity convexity boolean algebras logic and intangibles, sets and orderings, sets functions relations, sets of sets -filter topologies constructivism and choice nets and convergences.

Abstract: Sets and orderings: sets functions relations and orderings more about sups and infs sets of sets - filters topologies constructivism and choice nets and convergences. Algebra: elementary algebraic systems concrete categories the real numbers linearity convexity boolean algebras logic and intangibles. Topology and uniformity: toplogical spaces separation and regularity axioms compactness uniform spaces metric and uniform completeness Baire theory positive measure and integration. Topological vector spaces: norms normed operators generalized Riemann integrals Frechet derivatives metrization of groups and vector spaces barrels and other features of TVSs duality and weak compactness vector measures initial value problems.

322 citations