Prtop
... subconstruct of Prtop containing $ which is Cartesian closed. This result implies for instance that the full subconstruct FrPrtop of Prtop whose objects are the Fr6chet spaces is not Cartesian closed and that the subconstructs consisting of compact Hausdorff pretopological spaces or locally compact ...
... subconstruct of Prtop containing $ which is Cartesian closed. This result implies for instance that the full subconstruct FrPrtop of Prtop whose objects are the Fr6chet spaces is not Cartesian closed and that the subconstructs consisting of compact Hausdorff pretopological spaces or locally compact ...
Fibre Bundles and Homotopy Exact Sequence
... pairs f : (X, A) → (Y, B) is a continuous map f : X → Y such that f (A) ⊂ B and f (x0 ) = y0 . Two maps f1 , f2 : (X, A) → (Y, B) are homotopic if there is a continuous map F : (X × I, A × I) → (Y, B) such that F (x, 0) = f1 (x), F (x, 1) = f2 (x). Homotopy is an equivalence relation and it makes se ...
... pairs f : (X, A) → (Y, B) is a continuous map f : X → Y such that f (A) ⊂ B and f (x0 ) = y0 . Two maps f1 , f2 : (X, A) → (Y, B) are homotopic if there is a continuous map F : (X × I, A × I) → (Y, B) such that F (x, 0) = f1 (x), F (x, 1) = f2 (x). Homotopy is an equivalence relation and it makes se ...
1. Divisors Let X be a complete non-singular curve. Definition 1.1. A
... Definition 1.3. Let f ∈ k(X)∗ be a non-zero rational function on X. We define the P divisor of f to be the divisor div f = P ∈X vP (f ) · P on X. By the Lemma, this is well-defined. The map k(X)∗ → Z(X) given by f 7→ div f is a homomorphism of abelian groups. The image is called the group of princip ...
... Definition 1.3. Let f ∈ k(X)∗ be a non-zero rational function on X. We define the P divisor of f to be the divisor div f = P ∈X vP (f ) · P on X. By the Lemma, this is well-defined. The map k(X)∗ → Z(X) given by f 7→ div f is a homomorphism of abelian groups. The image is called the group of princip ...
Advanced Algebra I
... That is, there is an embedding σ̄ : E → L such that σ̄|K = σ. We remark that L is not necessarily an algebraic closure of K. For example, L could be something like K(x), an algebraic closure of K(x). In order to prove the uniqueness, we need the following useful Lemma. Sketch of the proof. The stari ...
... That is, there is an embedding σ̄ : E → L such that σ̄|K = σ. We remark that L is not necessarily an algebraic closure of K. For example, L could be something like K(x), an algebraic closure of K(x). In order to prove the uniqueness, we need the following useful Lemma. Sketch of the proof. The stari ...
7. Sheaves Definition 7.1. Let X be a topological space. A presheaf
... Definition 7.5. A ring R is called a local ring if there is a unique maximal ideal. Definition 7.6. Let X be a topological space and let F be a presheaf on X. If p ∈ X let I ⊂ Top(X) be the full subcategory of Top(X) whose objects are those open sets which contain p. The stalk of F at p, denoted Fp ...
... Definition 7.5. A ring R is called a local ring if there is a unique maximal ideal. Definition 7.6. Let X be a topological space and let F be a presheaf on X. If p ∈ X let I ⊂ Top(X) be the full subcategory of Top(X) whose objects are those open sets which contain p. The stalk of F at p, denoted Fp ...
Closed locally path-connected subspaces of finite
... X of x there is another neighborhood V ⊂ U of x such that each point y ∈ V can be connected with x by a subcontinuum K ⊂ U . A space X is locally continuum-connected if it is locally continuumconnected at each point. It is clear that X is locally continuumconnected at x ∈ X if X is locally path-conn ...
... X of x there is another neighborhood V ⊂ U of x such that each point y ∈ V can be connected with x by a subcontinuum K ⊂ U . A space X is locally continuum-connected if it is locally continuumconnected at each point. It is clear that X is locally continuumconnected at x ∈ X if X is locally path-conn ...
Very dense subsets of a topological space.
... X is very dense in X; that is, if X0 ,→ X is a quasi-homeomorphism. Proposition (10.3.2). — Let X be Jacobson, Z ⊆ X locally quasi-constructible. Then the subspace Z is Jacobson, and a point z ∈ Z is closed in Z iff it is closed in X. S Proposition (10.3.3). — Let X = α Uα be an open covering. Then ...
... X is very dense in X; that is, if X0 ,→ X is a quasi-homeomorphism. Proposition (10.3.2). — Let X be Jacobson, Z ⊆ X locally quasi-constructible. Then the subspace Z is Jacobson, and a point z ∈ Z is closed in Z iff it is closed in X. S Proposition (10.3.3). — Let X = α Uα be an open covering. Then ...
INEQUALITY APPROACH IN TOPOLOGICAL CATEGORIES
... and G converges to y for the final convergence f ξ if there exists F on X and x ∈ f −1 (y) such that x ∈ limξ F and G ≥f (F). A topological space can be considered as a particular convergence space by declaring that a filter converges to a point if it is finer than the neighborhood filter of the poi ...
... and G converges to y for the final convergence f ξ if there exists F on X and x ∈ f −1 (y) such that x ∈ limξ F and G ≥f (F). A topological space can be considered as a particular convergence space by declaring that a filter converges to a point if it is finer than the neighborhood filter of the poi ...
NOTES ON NON-ARCHIMEDEAN TOPOLOGICAL GROUPS
... be identified with a subgroup of Homeo (X) for some compact X and also with a subgroup of Is(M, d), topological group of isometries of some metric space (M, d) endowed with the pointwise topology (see also [34]). Similar characterizations are true for N A with compact zero-dimensional spaces X and u ...
... be identified with a subgroup of Homeo (X) for some compact X and also with a subgroup of Is(M, d), topological group of isometries of some metric space (M, d) endowed with the pointwise topology (see also [34]). Similar characterizations are true for N A with compact zero-dimensional spaces X and u ...
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.