General Topology
General Topology

Modal logics based on the derivative operation in topological spaces
Modal logics based on the derivative operation in topological spaces

General Topology I
General Topology I

TOPOLOGICAL GROUPS 1. Introduction Topological groups are
TOPOLOGICAL GROUPS 1. Introduction Topological groups are

PRODUCTIVE PROPERTIES IN TOPOLOGICAL GROUPS
PRODUCTIVE PROPERTIES IN TOPOLOGICAL GROUPS

... b) first countable ⇐⇒ metrizable (Birkhoff–Kakutani’s theorem); c) all σ-compact topological groups have countable cellularity (even more, they have the Knaster property). See [28] and [3, Corollary 5.4.8]); d) every topological group has a maximal group extension %G, called the Raı̆kov completion o ...
The greatest splitting topology and semiregularity
The greatest splitting topology and semiregularity

topologies on spaces of subsets
topologies on spaces of subsets

... 5.8). Typical among these is that the function a:zA(zA(X)) —»cvf(X), which maps a collection of sets into its union, is continuous (Theorems 5.7.1 and 5.7.2). Next we study the relationships between a function/:X —> Fand the function it induces among the hyperspaces (Theorem 5.10). We conclude this ...
The local structure of compactified Jacobians
The local structure of compactified Jacobians

LECTURE NOTES IN TOPOLOGICAL GROUPS 1
LECTURE NOTES IN TOPOLOGICAL GROUPS 1

Manifolds of smooth maps
Manifolds of smooth maps

ON DECOMPOSITION OF GENERALIZED CONTINUITY 1
ON DECOMPOSITION OF GENERALIZED CONTINUITY 1

ON WEAKLY e-CONTINUOUS FUNCTIONS
ON WEAKLY e-CONTINUOUS FUNCTIONS

A note on actions of a monoidal category
A note on actions of a monoidal category



ON θ-GENERALIZED CLOSED SETS
ON θ-GENERALIZED CLOSED SETS

Homotopy Theory of Topological Spaces and Simplicial Sets
Homotopy Theory of Topological Spaces and Simplicial Sets

ε-Open sets
ε-Open sets

Topologies on Spaces of Subsets Ernest Michael Transactions of
Topologies on Spaces of Subsets Ernest Michael Transactions of

Normal induced fuzzy topological spaces
Normal induced fuzzy topological spaces

... N (x0 ) ∩ {f (x) < rl} = φ. Thus, we infer that if there exists a point x0 ∈ X, f (x0 ) = r such that N (x0 ) is a subset of {x : f (x) > r} ∪ {x0 }, then the reverse inclusion is not true. Thus the property ∗ follows from the condition that the set of the form {x : f (x) < r}, which is a union of r ...
projective limits - University of California, Berkeley
projective limits - University of California, Berkeley

Topological Spaces and Continuous Functions
Topological Spaces and Continuous Functions

ON θ-PRECONTINUOUS FUNCTIONS
ON θ-PRECONTINUOUS FUNCTIONS

COMPLETION FUNCTORS FOR CAUCHY SPACES
COMPLETION FUNCTORS FOR CAUCHY SPACES

... denotes the filter generated on X by- (considered as a filter base on X). If (X, C) is a complete Cauchy space (i.e. convergence space), then it will be necessary to distinguish between a convergence subspace (a subspace in the usual convergence space sense) and a Cauchy subspace (with the meaning d ...
On qpI-Irresolute Mappings
On qpI-Irresolute Mappings

INTRODUCTION TO MANIFOLDS - PART 1/3 Contents 1. What is Algebraic Topology?
INTRODUCTION TO MANIFOLDS - PART 1/3 Contents 1. What is Algebraic Topology?

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They exist in several varieties such as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the ""usual"" topological cohomology theories such as singular cohomology. Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of differential equations. In addition, generalisations of sheaves to more general settings than topological spaces, such as Grothendieck topology, have provided applications to mathematical logic and number theory.