Properties of topological groups and Haar measure
... Definition 2.2. A Haar measure on a locally compact topological group G is a non–zero Radon measure which is right translation –invariant, i.e. µ(gE) = µ(E) for any Borel subset E of G and each g ∈ G. One can similarly define left translation–invariant measures, or bi-invariant translation –invarian ...
... Definition 2.2. A Haar measure on a locally compact topological group G is a non–zero Radon measure which is right translation –invariant, i.e. µ(gE) = µ(E) for any Borel subset E of G and each g ∈ G. One can similarly define left translation–invariant measures, or bi-invariant translation –invarian ...
A class of angelic sequential non-Fréchet–Urysohn topological groups
... non-locally compact dual group. The class C ∧ of the dual groups of members of C, is a family of angelic, sequential, hemicompact complete topological groups which are not Fréchet–Urysohn, while their compact subsets are even metrizable. In the same spirit, if V denotes the class formed by the infin ...
... non-locally compact dual group. The class C ∧ of the dual groups of members of C, is a family of angelic, sequential, hemicompact complete topological groups which are not Fréchet–Urysohn, while their compact subsets are even metrizable. In the same spirit, if V denotes the class formed by the infin ...
Multifunctions and graphs - Mathematical Sciences Publishers
... In [9], a function Φ: X —> Y is said to have a strongly-closed graph if for each (x, y) e (X x Y) — G{Φ) there are sets V e Σ(x) in X and WeΣ(y) in Γ with (Vx c\(W))Π G(Φ) = 0 . This notion is used in [9], [10], [11], and [12] to obtain characterizations of Hclosed and minimal Hausdorff spaces. If K ...
... In [9], a function Φ: X —> Y is said to have a strongly-closed graph if for each (x, y) e (X x Y) — G{Φ) there are sets V e Σ(x) in X and WeΣ(y) in Γ with (Vx c\(W))Π G(Φ) = 0 . This notion is used in [9], [10], [11], and [12] to obtain characterizations of Hclosed and minimal Hausdorff spaces. If K ...
A survey of ultraproduct constructions in general topology
... In the setting of concrete categories; i.e., those suitably endowed with forgetful functors to the category of sets and functions, the notion of ultrafiniteness can easily fail to coincide with that of having finite underlying set. For example, consider C = CH, the category of compacta (i.e., compac ...
... In the setting of concrete categories; i.e., those suitably endowed with forgetful functors to the category of sets and functions, the notion of ultrafiniteness can easily fail to coincide with that of having finite underlying set. For example, consider C = CH, the category of compacta (i.e., compac ...
Omega open sets in generalized topological spaces
... then (f |A )−1 (C) = f −1 (C) ∩ A is ω-(µ1 )A -closed in (A, (µ1 )A ) and (f |B )−1 (C) = f −1 (C) ∩ B is ω-(µ1 )B closed. By Lemma 3.8, it follows that (f |A )−1 (C) and (f |B )−1 (C) are ω-µ1 -closed in (X, µ1 ). It follows that f is ω-(µ1 , µ2 )-continuous. For any two generalized topological spa ...
... then (f |A )−1 (C) = f −1 (C) ∩ A is ω-(µ1 )A -closed in (A, (µ1 )A ) and (f |B )−1 (C) = f −1 (C) ∩ B is ω-(µ1 )B closed. By Lemma 3.8, it follows that (f |A )−1 (C) and (f |B )−1 (C) are ω-µ1 -closed in (X, µ1 ). It follows that f is ω-(µ1 , µ2 )-continuous. For any two generalized topological spa ...
Lecture Notes
... Proposition 4.3. If there is a continous map v : Sn → Sn such that v(x) ⊥ x for all x ∈ Sn , then the antpodal map is homotopic to the identity. The required homotopy is given by ht (x) := cos(πt)x + sin(πt)v(x) for t ∈ [0, 1]. For S1 note that this homotopy just rotates the circle through π radians ...
... Proposition 4.3. If there is a continous map v : Sn → Sn such that v(x) ⊥ x for all x ∈ Sn , then the antpodal map is homotopic to the identity. The required homotopy is given by ht (x) := cos(πt)x + sin(πt)v(x) for t ∈ [0, 1]. For S1 note that this homotopy just rotates the circle through π radians ...
Compact operators on Banach spaces
... This is the Fredholm alternative for operators T − λ with T compact and λ 6= 0: either T − λ is bijective, or has non-trivial kernel and non-trivial cokernel, of the same dimension. As above, the compactness of T implies the finite-dimensionality of ker(T − λ) for λ 6= 0. Dually, for y1 , . . . , yn ...
... This is the Fredholm alternative for operators T − λ with T compact and λ 6= 0: either T − λ is bijective, or has non-trivial kernel and non-trivial cokernel, of the same dimension. As above, the compactness of T implies the finite-dimensionality of ker(T − λ) for λ 6= 0. Dually, for y1 , . . . , yn ...
Free Topological Groups - Universidad Complutense de Madrid
... This is a good point to turn back to the problem of the existence of free topological groups. Let us consider the non-Abelian case first. It follows from Definition 1.1 that the topology of the group F (X ) (when the latter exists) is maximal in some sense. Here is the exact mathematical formulation ...
... This is a good point to turn back to the problem of the existence of free topological groups. Let us consider the non-Abelian case first. It follows from Definition 1.1 that the topology of the group F (X ) (when the latter exists) is maximal in some sense. Here is the exact mathematical formulation ...
Metrizability of hereditarily normal compact like groups1
... metrizable (this is [4, Corollary 2.5]), as well as item (d) of Example 1.3 (using Chaber’s theorem about the metrizabilty of the countably compact spaces with a G -diagonal). The result for item (a) can be deduced also from the above result and the fact, established by Nyikos, L. Soukup, B. Veličk ...
... metrizable (this is [4, Corollary 2.5]), as well as item (d) of Example 1.3 (using Chaber’s theorem about the metrizabilty of the countably compact spaces with a G -diagonal). The result for item (a) can be deduced also from the above result and the fact, established by Nyikos, L. Soukup, B. Veličk ...
Locally compact perfectly normal spaces may all be paracompact
... Our set-theoretic notation is standard, as in [17]. All ω1 -trees are presumed to be normal, in the terminology of [14]. If S is a tree and α is an ordinal, we let S(α) denote the αth level of S. Topological notation is from Engelking [8]. Since we mainly deal with locally compact spaces, it is conv ...
... Our set-theoretic notation is standard, as in [17]. All ω1 -trees are presumed to be normal, in the terminology of [14]. If S is a tree and α is an ordinal, we let S(α) denote the αth level of S. Topological notation is from Engelking [8]. Since we mainly deal with locally compact spaces, it is conv ...
The Main Conjecture - School of Mathematics, TIFR
... In his important papers [16], [17], Wiles proved in most cases the so-called main conjecture for the cyclotomic Zp -extension of any totally real base field F , for all odd primes p, and for all abelian characters of Gal(Q/F ) (we say most cases because his work only establishes the main conjecture ...
... In his important papers [16], [17], Wiles proved in most cases the so-called main conjecture for the cyclotomic Zp -extension of any totally real base field F , for all odd primes p, and for all abelian characters of Gal(Q/F ) (we say most cases because his work only establishes the main conjecture ...
II. General theory of locally compact groups
... are closed. We have shown that U ∩ V and U ∩ W are clopen. We have x ∈ S and S ⊂ (U ∩ V ) ∪ (U ∩ W ) , and thus x belongs either to the set U ∩ V or to the set U ∩ W . Assume that x ∈ U ∩ V . Then U ∩ V ∈ U and therefore S ⊂ U ∩ V . This, however, is a contradiction, since 6= N ⊂ S and V ∩ N =. Simi ...
... are closed. We have shown that U ∩ V and U ∩ W are clopen. We have x ∈ S and S ⊂ (U ∩ V ) ∪ (U ∩ W ) , and thus x belongs either to the set U ∩ V or to the set U ∩ W . Assume that x ∈ U ∩ V . Then U ∩ V ∈ U and therefore S ⊂ U ∩ V . This, however, is a contradiction, since 6= N ⊂ S and V ∩ N =. Simi ...
COUNTABLE DENSE HOMOGENEITY OF DEFINABLE SPACES 0
... either to 2ω or 2ω {0}, depending on whether it is compact or not. A natural question is whether the above results can be extended beyond analytic or Borel sets. The answer depends on set-theoretic assumptions. For possible extensions, note that all arguments presented so far use only the validity ...
... either to 2ω or 2ω {0}, depending on whether it is compact or not. A natural question is whether the above results can be extended beyond analytic or Borel sets. The answer depends on set-theoretic assumptions. For possible extensions, note that all arguments presented so far use only the validity ...
On Quasi Compact Spaces and Some Functions Key
... One of the fundamental ideas in all of mathematics is the notion of continuity. So much so that there has been a movement in recent years to categorize mathematics into two main parts, namely discrete mathematics and continuous mathematics. In topology there have been many variants of continuity con ...
... One of the fundamental ideas in all of mathematics is the notion of continuity. So much so that there has been a movement in recent years to categorize mathematics into two main parts, namely discrete mathematics and continuous mathematics. In topology there have been many variants of continuity con ...
topological closure of translation invariant preorders
... use an additional algebraic structure to this end. In particular, we focus on translation invariant preorders on topological groups and vector preorders on topological linear spaces. This property is naturally motivated from an algebraic viewpoint. Indeed, in the theory of partially ordered groups, ...
... use an additional algebraic structure to this end. In particular, we focus on translation invariant preorders on topological groups and vector preorders on topological linear spaces. This property is naturally motivated from an algebraic viewpoint. Indeed, in the theory of partially ordered groups, ...
1. Introduction and preliminaries
... Proof. Let {ValaEI} be a regular semiopen cover of Y. Then there is regular open Aa such that Aac VacCl(A a) for each aEI. Sincef is almost -continuous open, f -1 (Aa) Cf -1 (V a) ef- I (Cl (Aa» eCI (f -1 (Aa»' That is, {I-I(V a) laEI} is a semiopen cover of X so {j-I(Va) U Int(Cl(f- 1(Va» Ia E I} i ...
... Proof. Let {ValaEI} be a regular semiopen cover of Y. Then there is regular open Aa such that Aac VacCl(A a) for each aEI. Sincef is almost -continuous open, f -1 (Aa) Cf -1 (V a) ef- I (Cl (Aa» eCI (f -1 (Aa»' That is, {I-I(V a) laEI} is a semiopen cover of X so {j-I(Va) U Int(Cl(f- 1(Va» Ia E I} i ...
Fundamental Groups and Knots
... Now we have defined fundamental groups in a topological space, we are going to apply it to the study of knots and use it as an invariant for them. Definition: Two knots K1 and K2 contained in R3 are equivalent if there exists an orientationpreserving homeomorphism h: R3 → R3 such that h(K1) = K2. If ...
... Now we have defined fundamental groups in a topological space, we are going to apply it to the study of knots and use it as an invariant for them. Definition: Two knots K1 and K2 contained in R3 are equivalent if there exists an orientationpreserving homeomorphism h: R3 → R3 such that h(K1) = K2. If ...
THE EXACT SEQUENCE OF A SHAPE FIBRATION Q. Haxhibeqiri
... given in [5] we show when a restriction of shape fibration is again a shape fibration (Theorem 4.1) and when a shape fibration induces an isomorphism of homotopy pro-groups (Theorem 5.7) obtaining also the exaet sequence of shape fibration (Theorem 5.9). ...
... given in [5] we show when a restriction of shape fibration is again a shape fibration (Theorem 4.1) and when a shape fibration induces an isomorphism of homotopy pro-groups (Theorem 5.7) obtaining also the exaet sequence of shape fibration (Theorem 5.9). ...
The bordism version of the h
... on the construction of classifying Γ-spaces by Segal [41]. Here we introduce and use category valued partial Γ-sheaves in order to show that BMR is equivalent to BMR not only as a space, but also as an H -space with coherent operation. Similarly we construct a model BhMR for the classifying space of ...
... on the construction of classifying Γ-spaces by Segal [41]. Here we introduce and use category valued partial Γ-sheaves in order to show that BMR is equivalent to BMR not only as a space, but also as an H -space with coherent operation. Similarly we construct a model BhMR for the classifying space of ...
Full-Text PDF
... (i) A(G) is weakly compactly generated; (ii) C0 (G) is weakly compactly generated; (iii) G is first countable. In the sequel we shall several times use the following simple fact. Let E and F be Banach spaces, and suppose that E is WCG and that there exists a continuous linear mapping from E into F w ...
... (i) A(G) is weakly compactly generated; (ii) C0 (G) is weakly compactly generated; (iii) G is first countable. In the sequel we shall several times use the following simple fact. Let E and F be Banach spaces, and suppose that E is WCG and that there exists a continuous linear mapping from E into F w ...
some good extensions of compactness inšostak`s l-fuzzy
... [14]. We shall denote by LX the lattice of all L-fuzzy sets on X. For an ordinary subset A of X, we denote by χA the characteristic function of A. 2.1. Definition. [6]. An element p of L is called prime iff p 6= 1 and whenever a, b ∈ L with a ∧ b ≤ p then a ≤ p or b ≤ p. The set of all prime element ...
... [14]. We shall denote by LX the lattice of all L-fuzzy sets on X. For an ordinary subset A of X, we denote by χA the characteristic function of A. 2.1. Definition. [6]. An element p of L is called prime iff p 6= 1 and whenever a, b ∈ L with a ∧ b ≤ p then a ≤ p or b ≤ p. The set of all prime element ...
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.