SINGLE VALUED EXTENSION PROPERTY AND GENERALIZED WEYL’S THEOREM
... B-Fredholm operator of index 0. The B-Weyl spectrum σBW (T ) of T is defined by σBW (T ) = {λ ∈ : T − λI is not a B-Weyl operator}. If we consider a normal operator T acting on a Hilbert space H, Berkani proved in [4, Theorem 4.5] that σBW (T ) = σ(T ) \ E(T ), where E(T ) is the set of all isolated ...
... B-Fredholm operator of index 0. The B-Weyl spectrum σBW (T ) of T is defined by σBW (T ) = {λ ∈ : T − λI is not a B-Weyl operator}. If we consider a normal operator T acting on a Hilbert space H, Berkani proved in [4, Theorem 4.5] that σBW (T ) = σ(T ) \ E(T ), where E(T ) is the set of all isolated ...
INTERSECTION OF SETS WITH n
... convex sets in Rn has a nonempty intersection if (and only if ) the intersection of every n + 1 members of the family is nonempty. As is well known, Helly’s theorem can easily be proved by combining Klee’s theorem with Carathéodory’s theorem (see for example Berge [1, p. 173]). Recall that a subset ...
... convex sets in Rn has a nonempty intersection if (and only if ) the intersection of every n + 1 members of the family is nonempty. As is well known, Helly’s theorem can easily be proved by combining Klee’s theorem with Carathéodory’s theorem (see for example Berge [1, p. 173]). Recall that a subset ...
Symplectic Topology
... geometry is very special. For instance, R 2n has a unique smooth structure if n 6= 2, but R 4 has uncountably many. In the world of closed simply-connected manifolds, in a given homeomorphism type in dimension k 6= 4 there are at most finitely many diffeomorphism types – and (complicated) homotopy i ...
... geometry is very special. For instance, R 2n has a unique smooth structure if n 6= 2, but R 4 has uncountably many. In the world of closed simply-connected manifolds, in a given homeomorphism type in dimension k 6= 4 there are at most finitely many diffeomorphism types – and (complicated) homotopy i ...
Tychonoff`s Theorem
... Ea 6= ∅ for any a, then the axiom of choice says there is a function f : P → ∪a∈P Ea with f (a) ∈ Ea . This means that f (a) > a for every a. So, we create a chain a < f (a) < f (f (a)) < · · · . We know this chain has an upper bound since every chain in P has an upper bound. Call it b. Then f (b) ...
... Ea 6= ∅ for any a, then the axiom of choice says there is a function f : P → ∪a∈P Ea with f (a) ∈ Ea . This means that f (a) > a for every a. So, we create a chain a < f (a) < f (f (a)) < · · · . We know this chain has an upper bound since every chain in P has an upper bound. Call it b. Then f (b) ...
On topological groups via a-local functions - RiuNet
... where τ (x) = {U ∈ τ | x ∈ U }. A Kuratowski closure operator Cl∗ (.) for a topology τ ∗ (τ, I), called the ∗-topology, which is finer than τ is defined by Cl∗ (A) = A ∪ A∗ (τ, I), when there is no chance of confusion. A∗ (I) is denoted by A∗ and τ ∗ for τ ∗ (I, τ ). X ∗ is often a proper subset of ...
... where τ (x) = {U ∈ τ | x ∈ U }. A Kuratowski closure operator Cl∗ (.) for a topology τ ∗ (τ, I), called the ∗-topology, which is finer than τ is defined by Cl∗ (A) = A ∪ A∗ (τ, I), when there is no chance of confusion. A∗ (I) is denoted by A∗ and τ ∗ for τ ∗ (I, τ ). X ∗ is often a proper subset of ...
SOME FIXED-POINT THEOREMS ON AN ALMOST G
... Γ∗ − KKM mapping F with respect to T the family {F (x)|x ∈ X} has the finite intersection property, then T is said to have the Γ∗ −KKM property. The class Γ∗ − KKM (X, Y ) is defined to be the set {T : X → 2 Y |T has the Γ∗ − KKM property}. Lemma 1. Let X be an almost G-convex subset of a G-convex s ...
... Γ∗ − KKM mapping F with respect to T the family {F (x)|x ∈ X} has the finite intersection property, then T is said to have the Γ∗ −KKM property. The class Γ∗ − KKM (X, Y ) is defined to be the set {T : X → 2 Y |T has the Γ∗ − KKM property}. Lemma 1. Let X be an almost G-convex subset of a G-convex s ...
Note on the Tychonoff theorem and the axiom of choice.
... Then 0, 1, 2 are all cluster points of this net. Note that if we view the range of this net as a subset of R, then the limit points of this set are 0 and 1. Theorem 8. Let X be a topological space. The following are equivalent: (a) X is compact. (b) Every collection of closed subsets of X with the F ...
... Then 0, 1, 2 are all cluster points of this net. Note that if we view the range of this net as a subset of R, then the limit points of this set are 0 and 1. Theorem 8. Let X be a topological space. The following are equivalent: (a) X is compact. (b) Every collection of closed subsets of X with the F ...
Unified operation approach of generalized closed sets via
... Definition 2 A subset A of a topological space (X, τ ) is called (I, γ)-generalized closed if A∗ ⊆ U γ whenever A ⊆ U and U is open in (X, τ ). We denote the family of all (I, γ)-generalized closed subsets of a space (X, τ, I, γ) by IG(X) and simply write I-generalized closed (= I-g-closed) in case ...
... Definition 2 A subset A of a topological space (X, τ ) is called (I, γ)-generalized closed if A∗ ⊆ U γ whenever A ⊆ U and U is open in (X, τ ). We denote the family of all (I, γ)-generalized closed subsets of a space (X, τ, I, γ) by IG(X) and simply write I-generalized closed (= I-g-closed) in case ...
USC3002 Picturing the World Through Mathematics
... Theorem A 2nd countable space 1st countable and separable. Proof Assume that X is 2nd countable with a countable basis B. Then for every p X the set B p {O B : p O} is a local basis at p. For every nonempty O B choose xO O and define the set D {xO : (O B) (O )}. Then D is count ...
... Theorem A 2nd countable space 1st countable and separable. Proof Assume that X is 2nd countable with a countable basis B. Then for every p X the set B p {O B : p O} is a local basis at p. For every nonempty O B choose xO O and define the set D {xO : (O B) (O )}. Then D is count ...
This Ain`t No Meager Theorem - Department of Mathematics
... of open balls. It is easily seen that every metric space is a topological space, though not all topologies can be generated by a metric. A metric space is called complete if every Cauchy sequence converges, and a topological space is called topologically complete if it is homeomorphic to a complete ...
... of open balls. It is easily seen that every metric space is a topological space, though not all topologies can be generated by a metric. A metric space is called complete if every Cauchy sequence converges, and a topological space is called topologically complete if it is homeomorphic to a complete ...
on maps: continuous, closed, perfect, and with closed graph
... The following theorem 4(theorem 5), part (b) of which is a generalization (analogue) of theorem 5 of Piotrowski and Szymanski [3], gives sufficient conditions under which the converse of theorem (theorem 2) holds. THEOREM 4. Let f:X Y have closed graph. Then f is continuous if any one of the followi ...
... The following theorem 4(theorem 5), part (b) of which is a generalization (analogue) of theorem 5 of Piotrowski and Szymanski [3], gives sufficient conditions under which the converse of theorem (theorem 2) holds. THEOREM 4. Let f:X Y have closed graph. Then f is continuous if any one of the followi ...
Topology Proceedings METRIZABILITY OF TOPOLOGICAL
... a semigroup with a continuous semigroup operation such that ab = ac, ba = ca and b = c are equivalent for any a, b, c ∈ L. A question that can be traced to Abel and Lie, and was listed as the second half of Hilbert’s fifth problem, essentially asks whether a cancellative topological semigroup on a c ...
... a semigroup with a continuous semigroup operation such that ab = ac, ba = ca and b = c are equivalent for any a, b, c ∈ L. A question that can be traced to Abel and Lie, and was listed as the second half of Hilbert’s fifth problem, essentially asks whether a cancellative topological semigroup on a c ...
A Theorem on Remainders of Topological Groups
... MGU and MPGU, Moscow, Russian Federation University of Amsterdam, Amsterdam, the Netherlands December 16, 2015 Abstract It has been established in [7], [8], and [9] that a non-locally compact topological group G with a first-countable remainder can fail to be metrizable. On the other hand, it was sh ...
... MGU and MPGU, Moscow, Russian Federation University of Amsterdam, Amsterdam, the Netherlands December 16, 2015 Abstract It has been established in [7], [8], and [9] that a non-locally compact topological group G with a first-countable remainder can fail to be metrizable. On the other hand, it was sh ...
Investigation on Weak form of Generalized Closed sets in Ideal
... (i) If AI and BA, then BI (ii) If AI and BI, then ABI. Let (X, τ) be a topological space andIan ideal of subsets of X. An idealtopological space is a topological space (X, τ)with an idealIonXand is denoted by (X, τ, I). Definition 2.41,4 For a subset A of X, (i) A*(I, ) = {xX: UAI for eve ...
... (i) If AI and BA, then BI (ii) If AI and BI, then ABI. Let (X, τ) be a topological space andIan ideal of subsets of X. An idealtopological space is a topological space (X, τ)with an idealIonXand is denoted by (X, τ, I). Definition 2.41,4 For a subset A of X, (i) A*(I, ) = {xX: UAI for eve ...
Section 41. Paracompactness - Faculty
... Note. “Nicolas Bourbaki” is the name adopted by a group of French mathematicians in 1935. They chose the name in honor of a French general who played an important role in the Franco-Prussian War of 1870–71. The group was formed initially to write an analysis textbook. They wanted a book modeled afte ...
... Note. “Nicolas Bourbaki” is the name adopted by a group of French mathematicians in 1935. They chose the name in honor of a French general who played an important role in the Franco-Prussian War of 1870–71. The group was formed initially to write an analysis textbook. They wanted a book modeled afte ...
on a reflective subcategory of the category of all topological spaces
... eYf=F(f)ex for all/g Horn (X, Y). By a topological property there is meant a full subcategory of T which is closed under the formation of equivalent ( = homeomorphic) objects (full replete subcategory of 7"). A productive (resp. a closed hereditary) topological property is such that it is closed und ...
... eYf=F(f)ex for all/g Horn (X, Y). By a topological property there is meant a full subcategory of T which is closed under the formation of equivalent ( = homeomorphic) objects (full replete subcategory of 7"). A productive (resp. a closed hereditary) topological property is such that it is closed und ...
a decomposition of continuity
... In 1922 Blumberg[1] introduced the notion of a real valued function on Euclidean space being densely approached at a point in its domain. Continuous functions satisfy this condition at each point of their domains. This concept was generalized by Ptak[7] in 1958 who used the term ’nearly continuous’, ...
... In 1922 Blumberg[1] introduced the notion of a real valued function on Euclidean space being densely approached at a point in its domain. Continuous functions satisfy this condition at each point of their domains. This concept was generalized by Ptak[7] in 1958 who used the term ’nearly continuous’, ...
The Bryant--Ferry--Mio--Weinberger construction of generalized
... Theorem 1.4 Let X be a generalized n–manifold, n 5. Then X has a resolution if and only if I.X / D 1. Remark The integer I.X / is called the Quinn index of the generalized manifold X . Since the action of b L on L preserves the Z –sectors, arbitrary degree-one normal maps gW N ! X can be used to c ...
... Theorem 1.4 Let X be a generalized n–manifold, n 5. Then X has a resolution if and only if I.X / D 1. Remark The integer I.X / is called the Quinn index of the generalized manifold X . Since the action of b L on L preserves the Z –sectors, arbitrary degree-one normal maps gW N ! X can be used to c ...
APPLICATIONS OF THE TARSKI–KANTOROVITCH FIXED
... where cl denotes the closure operator. Again, as a by–product, we obtain here another new characterization of continuity (cf. Proposition 6 and Theorem 9). Section 5 deals with the family K(X) of all nonempty compact subsets of a topological space X, endowed with the inclusion ⊇. This time the condi ...
... where cl denotes the closure operator. Again, as a by–product, we obtain here another new characterization of continuity (cf. Proposition 6 and Theorem 9). Section 5 deals with the family K(X) of all nonempty compact subsets of a topological space X, endowed with the inclusion ⊇. This time the condi ...
A Definition of Boundary Values of Solutions of Partial Differential
... Helgason's conjecture: Any simultaneous eigenfunction of the invariant differential operators of a Riemannian symmetric space of non-compact type has a Poisson integral representation of a hyperfunction on its maximal boundary. There had been several affirmative results in some cases. But the method ...
... Helgason's conjecture: Any simultaneous eigenfunction of the invariant differential operators of a Riemannian symmetric space of non-compact type has a Poisson integral representation of a hyperfunction on its maximal boundary. There had been several affirmative results in some cases. But the method ...
Chapter 2: Manifolds
... Then any element of the group can be written as g(a) where a = (a1 , · · · , an ) . Since the composition of two elements of G must be another element of G, we can write g(a)g(b) = g(φ(a, b)) where φ = (φ1 , · · · , φn ) are n functions of a and b. Then for a Lie group, the functions φ are smooth (r ...
... Then any element of the group can be written as g(a) where a = (a1 , · · · , an ) . Since the composition of two elements of G must be another element of G, we can write g(a)g(b) = g(φ(a, b)) where φ = (φ1 , · · · , φn ) are n functions of a and b. Then for a Lie group, the functions φ are smooth (r ...
A REMARK ON VETRIVEL`S EXISTENCE THEOREM ON
... In 1996, Vetrivel [5] proved an existence theorem on Ky fan’s best approximant for multifunction with open inverse values in the setting of Hausdorff locally convex topological vector spaces. Theorem 1.1. Let M be a nonempty compact convex subset of Hausdorff locally convex topological space E. Supp ...
... In 1996, Vetrivel [5] proved an existence theorem on Ky fan’s best approximant for multifunction with open inverse values in the setting of Hausdorff locally convex topological vector spaces. Theorem 1.1. Let M be a nonempty compact convex subset of Hausdorff locally convex topological space E. Supp ...
Equivariant K-theory
... § 2. EQUIVARIANT K-THEORY Let X be a compact G-space. The set of isomorphism classes of G-vector bundles on X forms an abelian semigroup under ©. The associated abelian group is called K(^(X) : its elements are formal differences Eo—E^ of G-vector bundles on X, modulo the equivalence relation EO—EI= ...
... § 2. EQUIVARIANT K-THEORY Let X be a compact G-space. The set of isomorphism classes of G-vector bundles on X forms an abelian semigroup under ©. The associated abelian group is called K(^(X) : its elements are formal differences Eo—E^ of G-vector bundles on X, modulo the equivalence relation EO—EI= ...
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE, FMedSci FAA, HonFREng (born 22 April 1929) is a British mathematician specialising in geometry.Atiyah grew up in Sudan and Egypt and spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study. He has been president of the Royal Society (1990–1995), master of Trinity College, Cambridge (1990–1997), chancellor of the University of Leicester (1995–2005), and president of the Royal Society of Edinburgh (2005–2008). Since 1997, he has been an honorary professor at the University of Edinburgh.Atiyah's mathematical collaborators include Raoul Bott, Friedrich Hirzebruch and Isadore Singer, and his students include Graeme Segal, Nigel Hitchin and Simon Donaldson. Together with Hirzebruch, he laid the foundations for topological K-theory, an important tool in algebraic topology, which, informally speaking, describes ways in which spaces can be twisted. His best known result, the Atiyah–Singer index theorem, was proved with Singer in 1963 and is widely used in counting the number of independent solutions to differential equations. Some of his more recent work was inspired by theoretical physics, in particular instantons and monopoles, which are responsible for some subtle corrections in quantum field theory. He was awarded the Fields Medal in 1966, the Copley Medal in 1988, and the Abel Prize in 2004.