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Concept# Open set

Summary

In mathematics, an open set is a generalization of an open interval in the real line.
In a metric space (a set along with a distance defined between any two points), an open set is a set that, along with every point P, contains all points that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).
More generally, an open set is a member of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, every subset can be open (the discrete topology), or no subset can be open except the space itself and the empty set (the i

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Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological

Topology

In mathematics, topology (from the Greek words τόπος, and λόγος) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function.

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Related lectures (94)

Let f be an integrable function on RN, a a point in RN and B a complex number. If the mean value of f on the sphere of centre a and radius r tends to B when r tends to 0, we show that the Fourier integral at a of f is summable to B in Cesàro means of order λ > (N-1)/2. Let now U be a bounded open subset of RN whose boundary ∂U is a real analytic submanifold of RN with dimension N-1. We deduce from the preceding result that the Fourier integral at a of the indicator function of U is summable in Cesàro means of order λ > (N-1)/2 to 1 if a ∈ U, to 1/2 if a ∈ ∂U and to 0 if a ∉ U. We then show that if the function defined on ∂U by y → ‖ y - a ‖ has only a finite number of critical points, then we can take λ less or equal to (N-1)/2 ; more precisely, it suffices to have λ > (N-3)/2 + σ(a|∂U), where σ (a|∂U) < 0 is the maximum of the oscillatory indices associated to the critical points of y → ‖ y - a ‖ ; this generalizes results obtained by Pinsky, Taylor and Popov in 1997. Finally, writing μ∂U for the natural measure supported by ∂U, P(D) for a differential operator with constant coefficients of order m and b for a C∞ function on RN, we show that, if a is a point outside ∂U such that ‖ y - a ‖ has only a finite number of critical points on ∂U, the Fourier integral at a of the distribution P(D) bμ∂U is summable to 0 in Cesàro means of order λ > (N-1)/2 + m + σ (a|∂U) ; this generalizes a result obtained by Gonzàlez Vieli in 2002.

We use variational techniques to prove existence and nonexistence results for the following singular elliptic system: {div(vertical bar del u vertical bar(p-2)del u) = theta z(q)/u(1-0), u > 0 in Omega is an element of W-0(,1p) (Omega), -div(vertical bar del z vertical bar(p-2)del z) = qz(q-1)u(theta), z > 0 in Omega, z is an element of W-0(1,p) (Omega), where Omega is a bounded open set in RN (N >= 2), p > 1, q > (land 0 < 0 < 1.

In this paper we prove an existence result for the following singular elliptic system {z > 0 in Omega, z is an element of W-0(iota,p)(Omega) : -Delta(p)z = a(x)z(q-iota)u(theta) , u > 0 in Omega, u is an element of W-0(iota,p)(Omega) : -Delta(p)u = b(x)z(q)u(theta-iota) , where Omega is a bounded open set in R-N (N >= 2), -Delta(p) is the p-laplacian operator, a(x) and b(x) are suitable Lebesgue functions and q > 0, 0 < theta < 1, p > 1 are positive parameters satisfying suitable assumptions. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim