Homework Set 3 Solutions are due Monday, November 9th.
... corresponds to the map induced by inclusion F → F 0 . Problem 5. Deduce from the previous problem that if f : F → G is a morphism of sheaves of abelian groups on X, then Im(f ) is canonically isomorphic to the subsheaf F 0 of G, where F 0 (U ) consists of those s ∈ G(U ) such that for all x ∈ X, the ...
... corresponds to the map induced by inclusion F → F 0 . Problem 5. Deduce from the previous problem that if f : F → G is a morphism of sheaves of abelian groups on X, then Im(f ) is canonically isomorphic to the subsheaf F 0 of G, where F 0 (U ) consists of those s ∈ G(U ) such that for all x ∈ X, the ...
BORNOLOGICAL CONVERGENCES A. Lechicki, S. Levi, and A
... Let (X, d) be a metric space. For subsets C and D of X, the Hausdorff distance between C and D is given by h(C, D) = inf{ε > 0 : C ⊆ B(D, ε) and D ⊆ B(C, ε)}, where B(A, ε) is the ε-enlargement of the set A of radius ε. The Hausdorff disH tance induces a convergence H on the power set 2X by defining ...
... Let (X, d) be a metric space. For subsets C and D of X, the Hausdorff distance between C and D is given by h(C, D) = inf{ε > 0 : C ⊆ B(D, ε) and D ⊆ B(C, ε)}, where B(A, ε) is the ε-enlargement of the set A of radius ε. The Hausdorff disH tance induces a convergence H on the power set 2X by defining ...
spaces in which compact sets have countable local bases
... Let X = f(x, y) £ R2: y > OÍ. Define U by (Ma, b) = |(a, y) £ X: \y- b\< l/n\ ¡7n(a, 0) = |(x, y) £ X: y < l/n, U defines ...
... Let X = f(x, y) £ R2: y > OÍ. Define U by (Ma, b) = |(a, y) £ X: \y- b\< l/n\ ¡7n(a, 0) = |(x, y) £ X: y < l/n, U defines ...
Exercise Sheet 3
... Ob(C) the map FX,X 0 : MorC (X, X 0 ) → MorD (F X, F X 0 ) is bijective. It is called essentially surjective if for any Y ∈ Ob(D) there exists X ∈ Ob(C) such that FX ∼ = Y . Prove that F is part of an equivalence of categories if and only if it is fully faithful and essentially surjective. 3. Show t ...
... Ob(C) the map FX,X 0 : MorC (X, X 0 ) → MorD (F X, F X 0 ) is bijective. It is called essentially surjective if for any Y ∈ Ob(D) there exists X ∈ Ob(C) such that FX ∼ = Y . Prove that F is part of an equivalence of categories if and only if it is fully faithful and essentially surjective. 3. Show t ...
On Totally sg-Continuity, Strongly sg
... q. Then clearly f is totally sg-continuous, but not totally continuous. Definition 3.2 A function f : (X, τ) → (Y, σ) is said to be strongly semigeneralized-continuous (briefly strongly sg-continuous) if the inverse image of every subset of (Y, σ) is a sg-clopen subset of (X, τ). It is clear that st ...
... q. Then clearly f is totally sg-continuous, but not totally continuous. Definition 3.2 A function f : (X, τ) → (Y, σ) is said to be strongly semigeneralized-continuous (briefly strongly sg-continuous) if the inverse image of every subset of (Y, σ) is a sg-clopen subset of (X, τ). It is clear that st ...
1.1. Algebraic sets and the Zariski topology. We have said in the
... the Zariski topology are in a sense “very small”. It follows from this that any two nonempty open subsets of An have a non-empty intersection, which is also unfamiliar from the standard topology of real analysis. Example 1.1.11. Here is another example that shows that the Zariski topology is “unusua ...
... the Zariski topology are in a sense “very small”. It follows from this that any two nonempty open subsets of An have a non-empty intersection, which is also unfamiliar from the standard topology of real analysis. Example 1.1.11. Here is another example that shows that the Zariski topology is “unusua ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.