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Separation axiom
About: Separation axiom is a research topic. Over the lifetime, 989 publications have been published within this topic receiving 11811 citations.
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1,997 citations
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
TL;DR: In this article, the topology of the prospect space itself is removed, the previous axioms are weakened, an infinite number of sure prospects are allowed, and the existence of a measurable utility is established.
Abstract: : Previous treatments of this approach brought topological considerations of the prospects space into the axioms. In this paper considerations of the topology of the prospect space itself are removed, the previous axioms are weakened an infinite number of sure prospects are allowed. On the basis of these axioms the existence of a measurable utility is established.
687 citations
28 Aug 1991
TL;DR: A new formalism is presented to reason about topological relations based upon the nine intersections of boundaries, interiors, and complements between two objects that is applicable as a foundation for an algebra over topological Relations.
Abstract: A new formalism is presented to reason about topological relations. It is applicable as a foundation for an algebra over topological relations. The formalism is based upon the nine intersections of boundaries, interiors, and complements between two objects. Properties of topological relations are determined by analyzing the nine intersections to detect, for instance, symmetric topological relations and pairs of converse topological relations. Based upon the standard rules for the transitivity of set inclusion, the intersections of the composition of two binary topological relations are determined. These intersections are then matched with the intersections of the eight fundamental topological relations, giving an interpretation to the composition of topological relations.
650 citations
Book•
31 Dec 1932
TL;DR: Consequences of axioms 0, 1, 2, 3, 4, 5-1, 5_2, 6$ and $7$ Concerning topological equivalence and the introduction of distance Appendix Bibliography Glossary as mentioned in this paper.
Abstract: Consequences of axioms 0 and 1 Consequences of axioms 0, 1 and 2 Consequences of axioms 0, 1, 2, 3 and 5 Consequences of axioms 0 and 1-5 Upper semi-continuous collections Consequences of axioms $0, 1, 2, 3, 4, 5_1, 5_2, 6$ and $7$ Concerning topological equivalence and the introduction of distance Appendix Bibliography Glossary.
370 citations