1. Scheme A ringed space is a pair (X,OX), where X is a topological
... Example 1.1. Let X be any topological space and Z be the ring of integers with the discrete topology. The constant sheaf1 Z on X is a sheaf of rings on X. The pair (X, Z) is a ringed space. Example 1.2. Let X be a complex manifold and OX be the sheaf of holomorphic functions on X. Then (X, OX ) is a ...
... Example 1.1. Let X be any topological space and Z be the ring of integers with the discrete topology. The constant sheaf1 Z on X is a sheaf of rings on X. The pair (X, Z) is a ringed space. Example 1.2. Let X be a complex manifold and OX be the sheaf of holomorphic functions on X. Then (X, OX ) is a ...
Connectedness in fuzzy topology
... of papers have appeared in literature featuring the application of fuzzy sets to pattern recognition, decision problems, function approximation, system theory, fuzzy logic, fuzzy algorithms, fuzzy automata, fuzzy grammars, fuzzy languages, fuzzy algebras, fuzzy topology, etc. [2], [7]. In this note, ...
... of papers have appeared in literature featuring the application of fuzzy sets to pattern recognition, decision problems, function approximation, system theory, fuzzy logic, fuzzy algorithms, fuzzy automata, fuzzy grammars, fuzzy languages, fuzzy algebras, fuzzy topology, etc. [2], [7]. In this note, ...
Chapter 5
... The locally compact Hausdorff case is nicer. First of all the condition on A ⊂ C(X) to separate the closed subsets of X (which may be difficult to prove!) can be reduced to a local condition. Theorem 5.19. Let X be a Hausdorff paracompact space and A ⊂ C(X) closed under locally finite sums and under ...
... The locally compact Hausdorff case is nicer. First of all the condition on A ⊂ C(X) to separate the closed subsets of X (which may be difficult to prove!) can be reduced to a local condition. Theorem 5.19. Let X be a Hausdorff paracompact space and A ⊂ C(X) closed under locally finite sums and under ...
Proofs - Maths TCD
... Proof. Suppose X is Hausdorff and A ⊂ X is compact. To show that X − A is open, let x ∈ X − A be given. Then for each y ∈ A there exist disjoint open sets Uy , Vy such that x ∈ Uy and y ∈ Vy . Since the sets Vy form an open cover of A, finitely many of them cover A by compactness. Suppose that Vy1 , ...
... Proof. Suppose X is Hausdorff and A ⊂ X is compact. To show that X − A is open, let x ∈ X − A be given. Then for each y ∈ A there exist disjoint open sets Uy , Vy such that x ∈ Uy and y ∈ Vy . Since the sets Vy form an open cover of A, finitely many of them cover A by compactness. Suppose that Vy1 , ...
An Introduction to Topology
... using de Morgan’s Rules: ( i∈I Oi )c = i∈I Oic and ( i∈I Oi )c = i∈I Oic . Lemma 1.1.11. In a semimetric space (X, d), we have: (i) The sets Br (x), x ∈ X, r > 0, are open. (ii) The sets B≤r (x) := {y ∈ X : d(x, y) ≤ r} are closed. (iii) (X, d) is hausdorff if and only if d is a metric. Proof. (i) L ...
... using de Morgan’s Rules: ( i∈I Oi )c = i∈I Oic and ( i∈I Oi )c = i∈I Oic . Lemma 1.1.11. In a semimetric space (X, d), we have: (i) The sets Br (x), x ∈ X, r > 0, are open. (ii) The sets B≤r (x) := {y ∈ X : d(x, y) ≤ r} are closed. (iii) (X, d) is hausdorff if and only if d is a metric. Proof. (i) L ...
Preservations of so-metrizable spaces
... that a family P of subsets of a space X is called closure-preserving if P ′ = {P : P ∈ P ′ } for each P ′ ⊂ P; is called hereditarily ∪ closure-preserving if any family {H(P) : P ∈ P} is closure-preserving, where each H(P) ⊂ P ∈ P. P = {Pn : n ∈ N} is called σ-hereditarily closure-preserving if Pn i ...
... that a family P of subsets of a space X is called closure-preserving if P ′ = {P : P ∈ P ′ } for each P ′ ⊂ P; is called hereditarily ∪ closure-preserving if any family {H(P) : P ∈ P} is closure-preserving, where each H(P) ⊂ P ∈ P. P = {Pn : n ∈ N} is called σ-hereditarily closure-preserving if Pn i ...
PDF
... / F (some authors do not include this axiom in the definition of filter) • If F ∈ F and F ⊂ G ⊂ X then G ∈ F. The first two axioms can be replaced by one: • Any finite intersection of elements of F is an element of F. with the usual understanding that the intersection of an empty family of subsets o ...
... / F (some authors do not include this axiom in the definition of filter) • If F ∈ F and F ⊂ G ⊂ X then G ∈ F. The first two axioms can be replaced by one: • Any finite intersection of elements of F is an element of F. with the usual understanding that the intersection of an empty family of subsets o ...
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.