Completeness and quasi-completeness
... most important topological vector spaces are quasi-complete. In particular, the later theorem on GelfandPettis integrals will apply to all these. Here we define completeness and quasi-completeness for general topological vector spaces, and show that quasi-completeness is preserved by important const ...
... most important topological vector spaces are quasi-complete. In particular, the later theorem on GelfandPettis integrals will apply to all these. Here we define completeness and quasi-completeness for general topological vector spaces, and show that quasi-completeness is preserved by important const ...
FUNCTIONS AND BAIRE SPACES 1. Preliminaries Throughout
... Kragujevac Journal of Mathematics Volume 35 Number 1 (2011), Pages 5–12. ...
... Kragujevac Journal of Mathematics Volume 35 Number 1 (2011), Pages 5–12. ...
On Colimits in Various Categories of Manifolds
... So far we’ve shown that Mn f d is poorly behaved with respect to even very nice pushouts and coequalizers. A simple example to show it’s poorly behaved with respect to direct limits is the sequence R → R2 → R3 → . . . , where the colimit R∞ is not a manifold. The only colimits of interest left are t ...
... So far we’ve shown that Mn f d is poorly behaved with respect to even very nice pushouts and coequalizers. A simple example to show it’s poorly behaved with respect to direct limits is the sequence R → R2 → R3 → . . . , where the colimit R∞ is not a manifold. The only colimits of interest left are t ...
co-γ-Compact Generalized Topologies and c
... Aı ∈ c (gγ ) for ı ∈ I. Then we have the following posibilities: (a) If Aı ∈ c (τ ) for each ı ∈ I, then A ∈ c (τ ) ⊂ c (gγ ) since c (τ ) is a topology. (b) If Aı ∈ c (gγ ) − c (τ ) for each ı ∈ I, then Aı ∈ gγ − τ for each ı ∈ I. Thus A ∈ τ or A ∈ gγ − τ. If A ∈ τ, choose an ı ∈ I, then Aı ∈ gγ − ...
... Aı ∈ c (gγ ) for ı ∈ I. Then we have the following posibilities: (a) If Aı ∈ c (τ ) for each ı ∈ I, then A ∈ c (τ ) ⊂ c (gγ ) since c (τ ) is a topology. (b) If Aı ∈ c (gγ ) − c (τ ) for each ı ∈ I, then Aı ∈ gγ − τ for each ı ∈ I. Thus A ∈ τ or A ∈ gγ − τ. If A ∈ τ, choose an ı ∈ I, then Aı ∈ gγ − ...
g *-closed sets in ideal topological spaces
... Theorem 2.27. Let (X, τ , I) be an ideal topological space and A⊆X. Then the following are equivalent. 1. A is Ig⋆ -closed, 2. A∪(X−A∗ ) is Ig⋆ -closed, 3. A∗ −A is Ig⋆ -open. Proof. (1)⇒(2) Suppose A is Ig⋆ -closed. If U is any g-open set such that A∪(X−A∗ )⊆U, then X−U⊆ X−(A∪(X−A∗ ))=X∩(A∪(A∗ )c ) ...
... Theorem 2.27. Let (X, τ , I) be an ideal topological space and A⊆X. Then the following are equivalent. 1. A is Ig⋆ -closed, 2. A∪(X−A∗ ) is Ig⋆ -closed, 3. A∗ −A is Ig⋆ -open. Proof. (1)⇒(2) Suppose A is Ig⋆ -closed. If U is any g-open set such that A∪(X−A∗ )⊆U, then X−U⊆ X−(A∪(X−A∗ ))=X∩(A∪(A∗ )c ) ...
Notes from Craigfest - University of Melbourne
... from between the functors − ⊗ B and Hom(B, −). This means that if A and C are R-modules, then there is an ismorphism Hom(A ⊗ B, C) ' Hom(A, Hom(B, C)). 2. By analogy, (so we aren’t being too precise here), there is an adjunction between −∧A and M aps∗ (A, −), giving isomorphisms M aps∗ (X ∧ A, Y ) ' ...
... from between the functors − ⊗ B and Hom(B, −). This means that if A and C are R-modules, then there is an ismorphism Hom(A ⊗ B, C) ' Hom(A, Hom(B, C)). 2. By analogy, (so we aren’t being too precise here), there is an adjunction between −∧A and M aps∗ (A, −), giving isomorphisms M aps∗ (X ∧ A, Y ) ' ...
Math 525 More notes about compactness (sec 26
... open interval in each column). Thus (R, τu ) is homeomorphic to a subspace of L (indeed, to the first column), but L is not homeomorphic to a subspace of R, since second countability is hereditary. Also, L is not metrizable, since it contains a subspace homeomorphic to SΩ , which is not metrizable ( ...
... open interval in each column). Thus (R, τu ) is homeomorphic to a subspace of L (indeed, to the first column), but L is not homeomorphic to a subspace of R, since second countability is hereditary. Also, L is not metrizable, since it contains a subspace homeomorphic to SΩ , which is not metrizable ( ...
Equivariant cohomology and equivariant intersection theory
... deduce these criteria from new, more general results concerning (rational) smoothness at an “attractive” fixed 7point of a torus action; these results are closely related to recent work of Arabia [4]. The exposition follows the original five lectures, except in Section 5 which replaces a lecture on ...
... deduce these criteria from new, more general results concerning (rational) smoothness at an “attractive” fixed 7point of a torus action; these results are closely related to recent work of Arabia [4]. The exposition follows the original five lectures, except in Section 5 which replaces a lecture on ...
COMPACT SPACES, ELEMENTARY SUBMODELS, AND THE
... Let κ be an inaccessible cardinal smaller than the first 1-extendible. Take X to be the one-point compactification of the disjoint sum of Dγ , for γ < κ. Let M be an elementary submodel of a suitable H(θ) with the property that M ∩ κ is an ordinal less than κ, and such that all subsets of γ are in ...
... Let κ be an inaccessible cardinal smaller than the first 1-extendible. Take X to be the one-point compactification of the disjoint sum of Dγ , for γ < κ. Let M be an elementary submodel of a suitable H(θ) with the property that M ∩ κ is an ordinal less than κ, and such that all subsets of γ are in ...
ON SOMEWHAT β-CONTINUITY, SOMEWHAT β
... Theorem 3.4: A function f : (X, τ) → (Y, σ) is hardly β-open if and only if intβ(f–1(A)) = 0/ for each set A ⊂ Y having the property that intβ(A) = 0/ and A containing a nonempty closed set. Proof: Assume f is hardly β-open. Let A ⊂ Y such that intβ(A) = 0/ and let F be a nonempty closed set contain ...
... Theorem 3.4: A function f : (X, τ) → (Y, σ) is hardly β-open if and only if intβ(f–1(A)) = 0/ for each set A ⊂ Y having the property that intβ(A) = 0/ and A containing a nonempty closed set. Proof: Assume f is hardly β-open. Let A ⊂ Y such that intβ(A) = 0/ and let F be a nonempty closed set contain ...
MANIFOLDS, COHOMOLOGY, AND SHEAVES
... To understand these lectures, it is essential to know some point-set topology, as in [3, Appendix A], and to have a passing acquaintance with the exterior calculus of differential forms on a Euclidean space, as in [3, Sections 1–4]. To be consistent with Eduardo Cattani’s lectures at this summer sch ...
... To understand these lectures, it is essential to know some point-set topology, as in [3, Appendix A], and to have a passing acquaintance with the exterior calculus of differential forms on a Euclidean space, as in [3, Sections 1–4]. To be consistent with Eduardo Cattani’s lectures at this summer sch ...
Metrisability of Manifolds in Terms of Function Spaces
... Proof. Let K ⊂ U ⊂ X with K compact and U open. Use local compactness and regularity to find a finite collection {C1 , . . . , Cn } of compact subsets of U whose interiors cover K. As ∪ni=1 Ci is a compact subset of U there is N ∈ N such that ∪ni=1 Ci ⊂ N ⊂ U . Then K ⊂ N̊ ⊂ U . Lemma 5. Suppose th ...
... Proof. Let K ⊂ U ⊂ X with K compact and U open. Use local compactness and regularity to find a finite collection {C1 , . . . , Cn } of compact subsets of U whose interiors cover K. As ∪ni=1 Ci is a compact subset of U there is N ∈ N such that ∪ni=1 Ci ⊂ N ⊂ U . Then K ⊂ N̊ ⊂ U . Lemma 5. Suppose th ...
§5 Manifolds as topological spaces
... subspace of RN must satisfy some a priori topological conditions. The space RN has certain properties inherited by all its subspaces. Recall that a topological space is Hausdorff if for any two points x 6= y it is possible to find open sets Ux 3 x and Uy 3 y s.t. Ux ∩ Uy = ∅. Proposition 5.1. Any su ...
... subspace of RN must satisfy some a priori topological conditions. The space RN has certain properties inherited by all its subspaces. Recall that a topological space is Hausdorff if for any two points x 6= y it is possible to find open sets Ux 3 x and Uy 3 y s.t. Ux ∩ Uy = ∅. Proposition 5.1. Any su ...
For printing - Mathematical Sciences Publishers
... from Theorem 5.8 in [19], together with our Theorem 2.6. For non-trivial examples of the spaces hypothesized in 2.10, the reader is referred to [6]. We now turn to another aspect of ^-compactness. It follows from the corollary to Theorem 1 in [12], that every completely regular space has a maximal ^ ...
... from Theorem 5.8 in [19], together with our Theorem 2.6. For non-trivial examples of the spaces hypothesized in 2.10, the reader is referred to [6]. We now turn to another aspect of ^-compactness. It follows from the corollary to Theorem 1 in [12], that every completely regular space has a maximal ^ ...
Unwinding and integration on quotients
... Uniqueness of the integral is immediate when the dual V ∗ separates points, meaning that for v 6= v 0 in V there is λ ∈ V ∗ with λv 6= λv 0 . This separation property certainly holds for Hilbert spaces: the map λw = hw, v − v 0 i is a continuous linear functional and λ(v − v 0 ) 6= 0 gives λv 6= λv ...
... Uniqueness of the integral is immediate when the dual V ∗ separates points, meaning that for v 6= v 0 in V there is λ ∈ V ∗ with λv 6= λv 0 . This separation property certainly holds for Hilbert spaces: the map λw = hw, v − v 0 i is a continuous linear functional and λ(v − v 0 ) 6= 0 gives λv 6= λv ...
there exists a finite subset
... of X. Then the proof follows from Theorem 5. 5. Sufficiency. Suppose that A E RO (X) and it is RS-compact relative to X. Let {Va laE17} be a cover of A by regular semi-open sets of the subspace A. For each aE/7, there exists an OaERO(A) such that Oac VacClA(Oa)cClx(Oa)' Since AERO(X), we have OaERO( ...
... of X. Then the proof follows from Theorem 5. 5. Sufficiency. Suppose that A E RO (X) and it is RS-compact relative to X. Let {Va laE17} be a cover of A by regular semi-open sets of the subspace A. For each aE/7, there exists an OaERO(A) such that Oac VacClA(Oa)cClx(Oa)' Since AERO(X), we have OaERO( ...
rg\alpha-closed sets and rg\alpha
... be a regular α-open set of X such that B ⊂ U. Then A ⊂ U. Since A is a rgα-closed we have αcl(A) ⊂ U. Now αcl(B) ⊂ αcl (αcl(A)) = αcl(A) ⊂ U. Therefore B is rgα-closed set in X. Remark 2.3. The converse of the theorem 2.10. need not be true in general. Consider the topological space (X, τ ), where X ...
... be a regular α-open set of X such that B ⊂ U. Then A ⊂ U. Since A is a rgα-closed we have αcl(A) ⊂ U. Now αcl(B) ⊂ αcl (αcl(A)) = αcl(A) ⊂ U. Therefore B is rgα-closed set in X. Remark 2.3. The converse of the theorem 2.10. need not be true in general. Consider the topological space (X, τ ), where X ...
Chain Conditions of Horn and Tarski
... 4. Borel-generated examples 5. Martin’s axiom 6. Proper Forcing Axiom 7. Applications of MA and PFA 8. Chain conditions on compact spaces 9. A relativization of MA and PFA 10. The split interval ...
... 4. Borel-generated examples 5. Martin’s axiom 6. Proper Forcing Axiom 7. Applications of MA and PFA 8. Chain conditions on compact spaces 9. A relativization of MA and PFA 10. The split interval ...
THEORY OF FREDHOLM OPERATORS AND
... This theorem is due to Atiyah and Singer. It does not seem to be easy to translate all known proofs of the periodicity theorem of K-theory to K^ -theory, when 9W is of type II. E.g., the proof given by Atiyah and Singer in [4] is not easy to generalize (see in particular the proof of Proposition 3.5 ...
... This theorem is due to Atiyah and Singer. It does not seem to be easy to translate all known proofs of the periodicity theorem of K-theory to K^ -theory, when 9W is of type II. E.g., the proof given by Atiyah and Singer in [4] is not easy to generalize (see in particular the proof of Proposition 3.5 ...
Compact factors in finally compact products of topological spaces
... Recently, X. Caicedo, using deep logical and set theoretical methods, proved results similar to Theorem 1.1 with no need of separation axioms, but only for products with arbitrarily large numbers of factors. For example, Caicedo proved that if all powers of a space X are Lindelöf then X is compact. ...
... Recently, X. Caicedo, using deep logical and set theoretical methods, proved results similar to Theorem 1.1 with no need of separation axioms, but only for products with arbitrarily large numbers of factors. For example, Caicedo proved that if all powers of a space X are Lindelöf then X is compact. ...
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.