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... a star refinement which is an open cover, then every open cover has a σ-discrete open refinement which covers, has a finitary combinatorial counterpart in this language of symmetric operators. That is we show that despite the infinitary nature of discreteness the theorem of Stone-Michael can be prov ...
... a star refinement which is an open cover, then every open cover has a σ-discrete open refinement which covers, has a finitary combinatorial counterpart in this language of symmetric operators. That is we show that despite the infinitary nature of discreteness the theorem of Stone-Michael can be prov ...
Topological Extensions of Linearly Ordered Groups
... Corollary. Let G be a countable linearly ordered commutative locally compact topological group. Then every semigroup inverse + such that the map i :G ∋ x ( x, x)∈ B is a topology τB+ on BG G G ...
... Corollary. Let G be a countable linearly ordered commutative locally compact topological group. Then every semigroup inverse + such that the map i :G ∋ x ( x, x)∈ B is a topology τB+ on BG G G ...
Lindelo¨f spaces C(X) over topological groups - E
... Technical University of Valencia (Spain). This research was also supported by the Spanish Ministery of Education, projects MTM 2005-01182 for the first two named authors and BFM 2003-05878 for the last two named authors. These are FEDER projects, cofinanced by the European Community. ...
... Technical University of Valencia (Spain). This research was also supported by the Spanish Ministery of Education, projects MTM 2005-01182 for the first two named authors and BFM 2003-05878 for the last two named authors. These are FEDER projects, cofinanced by the European Community. ...
Compactness 1
... Proof. (⇒) Let ε > 0. Then there exist subsets A1 , A1 , . . . , An of X such that diam Ai < ε for all i ∈ {1, 2, . . . , n} and ∪ni=1 Ai = X. We may assume that each Ai is non-empty and choose xi ∈ Ai . If x ∈ X, then x ∈ Ai for some i and hence, d(x, xi ) ≤ diam(Ai ) < ε. This show that X = ∪ni=1 ...
... Proof. (⇒) Let ε > 0. Then there exist subsets A1 , A1 , . . . , An of X such that diam Ai < ε for all i ∈ {1, 2, . . . , n} and ∪ni=1 Ai = X. We may assume that each Ai is non-empty and choose xi ∈ Ai . If x ∈ X, then x ∈ Ai for some i and hence, d(x, xi ) ≤ diam(Ai ) < ε. This show that X = ∪ni=1 ...
CONFIGURATION SPACE INTEGRALS AND TAYLOR TOWERS
... knots. Let Km be the space of long knots in Rm or S m , m ≥ 3. To simplify notation, we will often set K = K3 when we wish to distinguish the case of classical knots from all others. At the heart of our results are Bott-Taubes configuration space integrals [6] which are used for producing cohomology ...
... knots. Let Km be the space of long knots in Rm or S m , m ≥ 3. To simplify notation, we will often set K = K3 when we wish to distinguish the case of classical knots from all others. At the heart of our results are Bott-Taubes configuration space integrals [6] which are used for producing cohomology ...
9. Sheaf Cohomology Definition 9.1. Let X be a topological space
... open subset. But this is not enough. One needs instead the slightly stronger condition that H i (UI , F) = 0. Theorem 9.3 (Leray). If X is a topological space and F is a sheaf of abelian groups and U is an open cover such that H i (UI , F) = 0, for all i > 0 and indices I, then in fact the natural m ...
... open subset. But this is not enough. One needs instead the slightly stronger condition that H i (UI , F) = 0. Theorem 9.3 (Leray). If X is a topological space and F is a sheaf of abelian groups and U is an open cover such that H i (UI , F) = 0, for all i > 0 and indices I, then in fact the natural m ...
o PAIRWISE LINDELOF SPACES
... En este articulo se definen espacios p-Lindelof y se estudian sus propiedades y relaciones ...
... En este articulo se definen espacios p-Lindelof y se estudian sus propiedades y relaciones ...
Some Types Of Compactness Via Ideal
... The purpose of the present paper is to introduce and study some types of compactness modulo an ideal called βI-compact and countably βI-compact spaces. Throughout we work with a topological space (X, τ) (or simply X), where no separation axioms are assumed. The usual notation Cl(A) for the closure a ...
... The purpose of the present paper is to introduce and study some types of compactness modulo an ideal called βI-compact and countably βI-compact spaces. Throughout we work with a topological space (X, τ) (or simply X), where no separation axioms are assumed. The usual notation Cl(A) for the closure a ...
Open Covers and Symmetric Operators
... We show that each open cover in a topological space gives rise naturally to a symmetric operator on the underlying set. If each open cover has a star refinement, then in a natural way the induced symmetric operator also admits star refinements. Surprisingly the existence of a σ-discrete refinement o ...
... We show that each open cover in a topological space gives rise naturally to a symmetric operator on the underlying set. If each open cover has a star refinement, then in a natural way the induced symmetric operator also admits star refinements. Surprisingly the existence of a σ-discrete refinement o ...
Compact groups and products of the unit interval
... The Cantor space B£° can be continuously mapped onto I, from ([7], §15-3). Hence (G) w G) = OMG) m a p s continuously onto ti < . Finally, using Tietze's theorem, there exists a continuous map from G onto iw(G). Remark 3 3 . We have shown in the proof of Theorem 3 1 that if X is a compact Hausdorff ...
... The Cantor space B£° can be continuously mapped onto I, from ([7], §15-3). Hence (G) w G) = OMG) m a p s continuously onto ti < . Finally, using Tietze's theorem, there exists a continuous map from G onto iw(G). Remark 3 3 . We have shown in the proof of Theorem 3 1 that if X is a compact Hausdorff ...
Pseudouniform topologies on C(X) given by ideals
... This section contains the basic definitions and notations that will be followed in this paper. All topological concepts that are not defined here should be understood as in T [2] except for the following: a uniformity U on a set X does not have to satisfy U = {(x, x) : x ∈ X}. Therefore, uniform spa ...
... This section contains the basic definitions and notations that will be followed in this paper. All topological concepts that are not defined here should be understood as in T [2] except for the following: a uniformity U on a set X does not have to satisfy U = {(x, x) : x ∈ X}. Therefore, uniform spa ...
On resolvable spaces and groups - EMIS Home
... property of being G -dense in some of their compacti cations. It is then natural to ask: 3. Is every weakly-pseudocompact group resolvable? The proof of Theorem 3.2 does not give -resolvability of G. ...
... property of being G -dense in some of their compacti cations. It is then natural to ask: 3. Is every weakly-pseudocompact group resolvable? The proof of Theorem 3.2 does not give -resolvability of G. ...
Section 26. Compact Sets - Faculty
... the bounded set in half and observing that one half or the other half must contain an infinite number of terms of the sequence, and then iterating this process. This is the same proof presented in Analysis 1 (see the notes for Section 2-3). In the 1860s, Karl Weierstrass (1815–1897)rediscovered this ...
... the bounded set in half and observing that one half or the other half must contain an infinite number of terms of the sequence, and then iterating this process. This is the same proof presented in Analysis 1 (see the notes for Section 2-3). In the 1860s, Karl Weierstrass (1815–1897)rediscovered this ...
![THE HIGHER HOMOTOPY GROUPS 1. Definitions Let I = [0,1] be](http://s1.studyres.com/store/data/001160219_1-57248e6267c8b98f385fc9eb1b30c0a7-300x300.png)
THE HIGHER HOMOTOPY GROUPS 1. Definitions Let I = [0,1] be
... It is easy to check that α · β is continuous: when t1 = 1/2 then α(1, t2 , ..., tn ) = p and β(0, t2 , ..., tn ) = p since (1, t2 , ..., tn ), (0, t2 , .., tn ) ∈ ∂I n . On the other hand, if (t1 , ..., tn ) ∈ ∂I n then the points (2t1 , t2 , ..., tn ) and (2t1 − 2, t2 , ..., tn ) are also in ∂I n a ...
... It is easy to check that α · β is continuous: when t1 = 1/2 then α(1, t2 , ..., tn ) = p and β(0, t2 , ..., tn ) = p since (1, t2 , ..., tn ), (0, t2 , .., tn ) ∈ ∂I n . On the other hand, if (t1 , ..., tn ) ∈ ∂I n then the points (2t1 , t2 , ..., tn ) and (2t1 − 2, t2 , ..., tn ) are also in ∂I n a ...
On acyclic and simply connected open manifolds - ICMC
... be an acyclic and simply connected open n-manifold and let X̄ = X ∪ {∞} be its onepoint compactification. If X is homeomorphic to Rn , then it is clear that X̄ is a manifold; indeed, in this case, X̄ is homeomorphic to S n . Now, suppose that X̄ is a manifold. Then, by Theorem 3.1, X̄ is a homotopy ...
... be an acyclic and simply connected open n-manifold and let X̄ = X ∪ {∞} be its onepoint compactification. If X is homeomorphic to Rn , then it is clear that X̄ is a manifold; indeed, in this case, X̄ is homeomorphic to S n . Now, suppose that X̄ is a manifold. Then, by Theorem 3.1, X̄ is a homotopy ...
Non-Hausdorff multifunction generalization of the Kelley
... (a) F * is closed in T Π «(X, Y). (b) F is pointwise bounded, and (c) F is evenly continuous. If X is a k-space and Y is regular, then the conditions (a), (b) and (c) are necessary for the compactness of F. Proof Sufficiency. Let F denote the τp -closure of F in Γ. Since T C (Y m X ) 0 , (c) implies ...
... (a) F * is closed in T Π «(X, Y). (b) F is pointwise bounded, and (c) F is evenly continuous. If X is a k-space and Y is regular, then the conditions (a), (b) and (c) are necessary for the compactness of F. Proof Sufficiency. Let F denote the τp -closure of F in Γ. Since T C (Y m X ) 0 , (c) implies ...
Existence of Maximal Elements and Equilibria in Linear Topological
... It should be emphasized that the method of proof given in BorglinKeiding (1976, p. 315) cannot be carried out to allow for an infinite number of commodities and a countably infinite number of agents. In particular, it fails due to the fact that the countably infinite intersection of open sets in a l ...
... It should be emphasized that the method of proof given in BorglinKeiding (1976, p. 315) cannot be carried out to allow for an infinite number of commodities and a countably infinite number of agents. In particular, it fails due to the fact that the countably infinite intersection of open sets in a l ...
RADON-NIKOD´YM COMPACT SPACES OF LOW WEIGHT AND
... Theorem 3. If K is a quasi Radon-Nikodým compact space of weight less than b, then K is Radon-Nikodým compact. The weight of a topological space is the least cardinality of a base for its topology. We also recall the definition of cardinal b. In the set NN we consider the order relation given by σ ...
... Theorem 3. If K is a quasi Radon-Nikodým compact space of weight less than b, then K is Radon-Nikodým compact. The weight of a topological space is the least cardinality of a base for its topology. We also recall the definition of cardinal b. In the set NN we consider the order relation given by σ ...
Since Lie groups are topological groups (and manifolds), it is useful
... there is some open subset, U , of G so that U \ H = {h}. Proposition 2.15. If G is a topological group and H is discrete subgroup of G, then H is closed. Proposition 2.16. If G is a topological group and H is any subgroup of G, then the closure, H, of H is a subgroup of G. Proposition 2.17. Let G be ...
... there is some open subset, U , of G so that U \ H = {h}. Proposition 2.15. If G is a topological group and H is discrete subgroup of G, then H is closed. Proposition 2.16. If G is a topological group and H is any subgroup of G, then the closure, H, of H is a subgroup of G. Proposition 2.17. Let G be ...
Section 29. Local Compactness - Faculty
... a compact subspace of Rω that contains x ∈ Rω and there is a neighborhood of x in C, then the neighborhood contains a basis element of the form of B. But then B = [a1 , b1 ] × [a1 , a2] × · · · × [an , bn ] × R × R × · · · is a closed subspace of C and so would be compact by Theorem 26.2. But B is n ...
... a compact subspace of Rω that contains x ∈ Rω and there is a neighborhood of x in C, then the neighborhood contains a basis element of the form of B. But then B = [a1 , b1 ] × [a1 , a2] × · · · × [an , bn ] × R × R × · · · is a closed subspace of C and so would be compact by Theorem 26.2. But B is n ...
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.