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normal and I g - Italian Journal of Pure and Applied Mathematics
normal and I g - Italian Journal of Pure and Applied Mathematics

Topology I with a categorical perspective
Topology I with a categorical perspective

ON THE UPPER LOWER SUPER. D-CONTINUOUS
ON THE UPPER LOWER SUPER. D-CONTINUOUS

Orbifolds and their cohomology.
Orbifolds and their cohomology.

“Research Note” TOPOLOGICAL RING
“Research Note” TOPOLOGICAL RING

... A topological groupoid is a groupoid R such that the sets R and R0 are topological spaces, and source, target, object, inverse and composition maps are continuous. Let R and H be two topological groupoids. A morphism of topological groupoids is a pair of maps f:H→R and f0:H0→R0 such that f and f0 ar ...
Pre-Semi-Closed Sets and Pre-Semi
Pre-Semi-Closed Sets and Pre-Semi

On some locally closed sets and spaces in Ideal Topological
On some locally closed sets and spaces in Ideal Topological

... (ii) A = Ucl(A) for some δ̂ s - open set U. (iii) cl(A) – A is δ̂ s - closed. (iv) A(X–cl(A)) is δ̂ s - open. Proof. (i)(ii) If A δ̂ sILC, then there exist a δ̂ s – open set U and a -I-closed set F such that A = UF. Clearly AUcl(A). Since F is -I-closed, cl(A)  cl(F) = F and so Uc ...
bases. Sub-bases. - Dartmouth Math Home
bases. Sub-bases. - Dartmouth Math Home

Vasile Alecsandri” University of Bac˘au Faculty of Sciences Scientific
Vasile Alecsandri” University of Bac˘au Faculty of Sciences Scientific

Full Text
Full Text

... For any point x of a topological space (X, τ ), τ (x) denotes the collection of all open neighborhoods of x. 2.1. Definition. [14] Let (X, τ ) be a topological space and G be a grill on X. A mapping Φ : P(X) → P(X) is defined as follows: Φ(A) = ΦG (A, τ ) = {x ∈ X : A ∩ U ∈ G for all U ∈ τ (x)} for ...
Universal covering spaces and fundamental groups in
Universal covering spaces and fundamental groups in

Print this article - Innovative Journal
Print this article - Innovative Journal

... A function f : (X, τ) → (Y, σ) is said to be semicontinuous[9] (resp. α-continuous [12], pre-continuous [11], totally continuous [7], totally semi-continuous [16]) if the inverse image of every open subset of (Y, σ) is a semi-open (resp. α-open, preopen, clopen, semi-clopen) subset of (X,τ). Definit ...
FURTHER DECOMPOSITIONS OF ∗-CONTINUITYI 1 Introduction
FURTHER DECOMPOSITIONS OF ∗-CONTINUITYI 1 Introduction

- Journal of Linear and Topological Algebra
- Journal of Linear and Topological Algebra

... open sets [16] and semi-preopen sets [13] . Multifunctions and of course continuous multifunctions stand among the most important and most researched points in the whole of the mathematical science. Many different forms of continuous multifunctions have been introduced over the years. Csaszar [1] in ...
Topology 550A Homework 3, Week 3 (Corrections
Topology 550A Homework 3, Week 3 (Corrections

... Proof . Consider, for α ∈ A, U = α Gα s.t. Gα are the basic neighborhoods, that is, the usual open disks centered at x that lie above the x-axis. It follows that U in in Γ. When we take the closure of U, ...
Pdf file
Pdf file

On a Simultaneous Generalization of β-Normality and - PMF-a
On a Simultaneous Generalization of β-Normality and - PMF-a

FASCICULI MATHEMAT ICI
FASCICULI MATHEMAT ICI

Topology Proceedings
Topology Proceedings

ESAKIA SPACES VIA IDEMPOTENT SPLIT COMPLETION
ESAKIA SPACES VIA IDEMPOTENT SPLIT COMPLETION

On Maps and Generalized Λb-Sets
On Maps and Generalized Λb-Sets

... The converse needs not be true as seen from the following example. Example 3.5. Let X = Y = {a, b, c}, τ = {∅, {a}, X} and σ = {∅, {a, b}, Y }. The identity map f : (X, τ ) → (Y, σ) is g.Λb-continuous but is not g.Λb -irresolute since for the g.Λb -set {b} of (Y, σ), the inverse image f −1 ({b}) = { ...
Basic Algebraic Geometry
Basic Algebraic Geometry

PDF file without embedded fonts
PDF file without embedded fonts

... begin with two disjoint subsets C; D  R and for each x 2 D a sequence hxn i in C converging to x. They let X(C; D) be the union C [ D but with points of C isolated and neighbourhoods of points of D containing tails of the corresponding sequences. The essential features of X(C; D) are then preserved ...
On Kolmogorov Topological Spaces 1
On Kolmogorov Topological Spaces 1

covariant and contravariant approaches to topology
covariant and contravariant approaches to topology

< 1 ... 18 19 20 21 22 23 24 25 26 ... 66 >

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.
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