Topological and Limit-space Subcategories of Countably
... if there exists some countable base for its topology (Smyth, 1992). Such spaces are also known as second-countable spaces. We write ωTop for the category of countably based topological spaces and ωEqu for the category of those equilogical spaces (X, ∼) where X is countably based. As mentioned in the ...
... if there exists some countable base for its topology (Smyth, 1992). Such spaces are also known as second-countable spaces. We write ωTop for the category of countably based topological spaces and ωEqu for the category of those equilogical spaces (X, ∼) where X is countably based. As mentioned in the ...
Homotopy theories and model categories
... and weak equivalences, which satisfy a few simple axioms that are deliberately reminiscent of properties of topological spaces. Surprisingly enough, these axioms give a reasonably general context in which it is possible to set up the basic machinery of homotopy theory. The machinery can then be used ...
... and weak equivalences, which satisfy a few simple axioms that are deliberately reminiscent of properties of topological spaces. Surprisingly enough, these axioms give a reasonably general context in which it is possible to set up the basic machinery of homotopy theory. The machinery can then be used ...
Covering property - Dipartimento di Matematica Tor Vergata
... [B, A]-compactness is what is usually called an irreducible (or minimal ) cover. Irreducible covers, as well as spaces in which every cover can be refined to a (possibly finite) irreducible cover have been the object of some study. See [2, 15] and further references there. In a sense, an infinite ir ...
... [B, A]-compactness is what is usually called an irreducible (or minimal ) cover. Irreducible covers, as well as spaces in which every cover can be refined to a (possibly finite) irreducible cover have been the object of some study. See [2, 15] and further references there. In a sense, an infinite ir ...
Notes
... surjects onto g as a Lie algebra with kernel a Lie bialgebra ideal. Thus, g inherits a quasitriangular Lie bialgebra structure from the double D(b). 3. Hopf algebras. Given G a Lie group, consider C ∞ (G) and U g = (C ∞ (G))∗ . The Lie structure on G gives rise to extra structure on these function s ...
... surjects onto g as a Lie algebra with kernel a Lie bialgebra ideal. Thus, g inherits a quasitriangular Lie bialgebra structure from the double D(b). 3. Hopf algebras. Given G a Lie group, consider C ∞ (G) and U g = (C ∞ (G))∗ . The Lie structure on G gives rise to extra structure on these function s ...
Zero-pointed manifolds
... In this work, we introduce zero-pointed manifolds as a tool to solve two apparently separate problems. The first problem, from manifold topology, is to generalize Poincaré duality to factorization homology; the second problem, from algebra, is to show the Koszul self-duality of n-disk, or En , alge ...
... In this work, we introduce zero-pointed manifolds as a tool to solve two apparently separate problems. The first problem, from manifold topology, is to generalize Poincaré duality to factorization homology; the second problem, from algebra, is to show the Koszul self-duality of n-disk, or En , alge ...
HYPERELLIPTIC JACOBIANS AND SIMPLE GROUPS U3 1
... when Rf can be identified with the corresponding “Hermitian curve” of isotropic lines in the projective plane P2 (Fq2 ) in such a way that Gal(f ) becomes either the projective unitary group PGU3 (Fq ) or the projective special unitary group U3 (q) := PSU3 (Fq ). In this case n = deg(f ) = q 3 + 1 = ...
... when Rf can be identified with the corresponding “Hermitian curve” of isotropic lines in the projective plane P2 (Fq2 ) in such a way that Gal(f ) becomes either the projective unitary group PGU3 (Fq ) or the projective special unitary group U3 (q) := PSU3 (Fq ). In this case n = deg(f ) = q 3 + 1 = ...
NOTES ON THE SEPARABILITY OF C*-ALGEBRAS Chun
... Let P (A) be the pure state space of A. In the abelian case, P (C0 (Ω)) ∼ = Ω. In general, we set Q(A) = {ϕ ∈ UA∗ : ϕ ≥ 0} to be the quasi-state space of A. Then Q(A) is a weak* compact convex set with extreme boundary P (A) ∪ {0}. Recall that a (closed) ideal I of a C*-algebra A is primitive if it ...
... Let P (A) be the pure state space of A. In the abelian case, P (C0 (Ω)) ∼ = Ω. In general, we set Q(A) = {ϕ ∈ UA∗ : ϕ ≥ 0} to be the quasi-state space of A. Then Q(A) is a weak* compact convex set with extreme boundary P (A) ∪ {0}. Recall that a (closed) ideal I of a C*-algebra A is primitive if it ...
MATH 436 Notes: Homomorphisms.
... and f1 = 1Z it follows that θ is a homomorphism of monoids. It is trivial to check that it is a bijection and so induces an isomorphism between (Z, ·) and End((Z, +)). This completes the proof. The following is an important concept for homomorphisms: Definition 1.11. If f : G → H is a homomorphism o ...
... and f1 = 1Z it follows that θ is a homomorphism of monoids. It is trivial to check that it is a bijection and so induces an isomorphism between (Z, ·) and End((Z, +)). This completes the proof. The following is an important concept for homomorphisms: Definition 1.11. If f : G → H is a homomorphism o ...
STRUCTURE THEOREMS OVER POLYNOMIAL RINGS 1
... G, one of the other conditions is satisfied for some ring R then it is also satisfied by any other ring R satisfying the hypotheses of the theorem. We thank the referee for the elegant proofs of 3.3 and 8.1. 2. Change of Category We want to move to a category in which all of the indecomposable kG-mo ...
... G, one of the other conditions is satisfied for some ring R then it is also satisfied by any other ring R satisfying the hypotheses of the theorem. We thank the referee for the elegant proofs of 3.3 and 8.1. 2. Change of Category We want to move to a category in which all of the indecomposable kG-mo ...
Topological Groups Part III, Spring 2008
... group for each α ∈ A. Their complete direct product is G = α Gα equipped with the usual product topology τG and with multiplication given by (x ×G y)α = xα ×α yα . Definition 4.2. Suppose A is non-empty and (Gα , ×α ) is a group Q for each α ∈ A. Their direct product is the subgroup of the group G = ...
... group for each α ∈ A. Their complete direct product is G = α Gα equipped with the usual product topology τG and with multiplication given by (x ×G y)α = xα ×α yα . Definition 4.2. Suppose A is non-empty and (Gα , ×α ) is a group Q for each α ∈ A. Their direct product is the subgroup of the group G = ...
Finite topological spaces
... Introduction Definition and basic properties Some elements on the classification T0 -spaces and simplicial complexes Conclusion ...
... Introduction Definition and basic properties Some elements on the classification T0 -spaces and simplicial complexes Conclusion ...
On locally compact totally disconnected Abelian groups and their
... since a divisible subgroup of a discrete group is a direct factor. Hence ZP^((ZP, g), g)^(p~xgι, g)^{v°°, g)x(9u 9) which is a contradiction as observed above. Hence every neighborhood of g must contain a compact nonopen subgroup. This theorem shows that a reasonable conjecture for a possible auxili ...
... since a divisible subgroup of a discrete group is a direct factor. Hence ZP^((ZP, g), g)^(p~xgι, g)^{v°°, g)x(9u 9) which is a contradiction as observed above. Hence every neighborhood of g must contain a compact nonopen subgroup. This theorem shows that a reasonable conjecture for a possible auxili ...
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.