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Set Theory and General Topology
Published in Kenneth Kuttler, Modern Analysis, 2017
14. ↑ Let (Ω, τ) be a locally compact Hausdorff space with ℬ the basis of sets whose closures are compact and consider the set Ω ⋃ {∞} where ∞ is called the point at ∞ and is an element of Hc for all sets, H, compact in (Ω, τ). (This is called the one point compactification of a locally compact Hausdorff space.) Thus ∞ is a new point not in Ω. Show ℬ∪ {Hc : H is a compact set in (Ω, τ)} is a basis for a topology which makes Ω ∪ {∞} into a compact Hausdorff space and is therefore a normal space. Use this to show that if K is compact in (Ω, τ) and V is open in (Ω, τ) containing K, then there exists an open set W whose closure is compact such that
PRELIMINARIES
Published in Marko Kostić, Abstract Volterra Integro-Differential Equations, 2015
In the remaining part of this section, we shall recall the basic facts and definitions from the theory of integration of functions with values in locally convex spaces. By Q we denote a locally compact and separable metric space (for example, Q can be chosen to be a (bounded or unbounded) segment I in Rn, where n e {1, 2}) and by we denote a locally finite Borel measure defined on Q.
Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
A topological space is said to be locally compact if each member of the space has a compact neighborhood. For example, discrete spaces and Euclidean spaces are locally compact. We shall see in Chapter that no infinite dimensional normed space can be locally compact.
Locally Starplus-Compactness in L-Topological Spaces
Published in Fuzzy Information and Engineering, 2018
The class of locally compact spaces is far more wider than the class of compact spaces. The locally compact spaces often arise in topology and applications of topology to geometry, analysis and algebra. For example, the study of locally compact abelian group forms the foundation of harmonic analysis. It is well known that every compact space is locally compact but the converse need not be true. For example, the Euclidean space is locally compact but not compact. Topological manifolds share the local properties of Euclidean space and hence are locally compact. A locally compact space can be imbedded in a compact space, which is its compactification. One of the simplest compapctification of a space is the one point compactification, wherein one simply adjoins one new point to the space. The classical example of one point compactification is the embedding of the Gaussian plane of complex numbers into the Riemann sphere. The category of locally compact spaces has been applied in almost every subdiscipline of mathematics and hence it is important to formulate an appropriate version of local compactness in the L-fuzzy setting.