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Topological space
About: Topological space is a research topic. Over the lifetime, 10881 publications have been published within this topic receiving 154419 citations.
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08 Feb 2014TL;DR: In this paper, the authors present the most basic results on topological vector spaces, including the uniformity of vector spaces over non-discrete valuated fields, and the notion of boundedness of these fields.
Abstract: This chapter presents the most basic results on topological vector spaces. With the exception of the last section, the scalar field over which vector spaces are defined can be an arbitrary, non-discrete valuated field K; K is endowed with the uniformity derived from its absolute value. The purpose of this generality is to clearly identify those properties of the commonly used real and complex number field that are essential for these basic results. Section 1 discusses the description of vector space topologies in terms of neighborhood bases of 0, and the uniformity associated with such a topology. Section 2 gives some means for constructing new topological vector spaces from given ones. The standard tools used in working with spaces of finite dimension are collected in Section 3, which is followed by a brief discussion of affine subspaces and hyperplanes (Section 4). Section 5 studies the extremely important notion of boundedness. Metrizability is treated in Section 6. This notion, although not overly important for the general theory, deserves special attention for several reasons; among them are its connection with category, its role in applications in analysis, and its role in the history of the subject (cf. Banach [1]). Restricting K to subfields of the complex numbers, Section 7 discusses the transition from real to complex fields and vice versa.
4,183 citations
1,997 citations
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
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