Countable dense homogeneity and λ-sets
Countable dense homogeneity and λ-sets

METRIC TOPOLOGY: A FIRST COURSE
METRIC TOPOLOGY: A FIRST COURSE

... metric space and continuous function. This follows a familiar pattern in modern pure mathematics: one studies certain structured sets, along with “structurerespecting” functions between them. For example, in linear algebra the basic concepts are those of vector space and linear transformation. In th ...
Introduction to Topological Groups
Introduction to Topological Groups

Smooth manifolds - IME-USP
Smooth manifolds - IME-USP

Open problems on countable dense homogeneity
Open problems on countable dense homogeneity

The Hilbert–Smith conjecture for three-manifolds
The Hilbert–Smith conjecture for three-manifolds

Thom Spectra that Are Symmetric Spectra
Thom Spectra that Are Symmetric Spectra

Modern descriptive set theory
Modern descriptive set theory

Elementary Topology Problem Textbook O. Ya. Viro, O. A. Ivanov, N
Elementary Topology Problem Textbook O. Ya. Viro, O. A. Ivanov, N

On Fuzzy Topological Spaces Involving Boolean Algebraic Structures
On Fuzzy Topological Spaces Involving Boolean Algebraic Structures

Topological Subset Space Models for Public
Topological Subset Space Models for Public

Equivariant K-theory
Equivariant K-theory

Paracompactness with respect to an ideal
Paracompactness with respect to an ideal

MAXIMAL QHC-SPACES 1. Introduction and Background. In 1924
MAXIMAL QHC-SPACES 1. Introduction and Background. In 1924

on if generalized* minimal open set
on if generalized* minimal open set

first four chapters - Jesse Johnson`s Website
first four chapters - Jesse Johnson`s Website

Simplicial Sets - Stanford Computer Graphics
Simplicial Sets - Stanford Computer Graphics

CHARACTERIZING CONTINUITY BY PRESERVING
CHARACTERIZING CONTINUITY BY PRESERVING

Derived algebraic geometry
Derived algebraic geometry

Introduction to Topological Spaces and Set-Valued Maps
Introduction to Topological Spaces and Set-Valued Maps

Covariance algebra of a partial dynamical system - MATH Mail
Covariance algebra of a partial dynamical system - MATH Mail

... (here (π, U, H) and (π ′ , U ′ , H ′) denote covariant representations of (A, δ)). In many cases the main interest is concentrated on the subcategory CovFaith(A, δ) of Cov(A, δ) for which objects are algebras C ∗ (π(A), U) where now π is faithful. The fundamental problem then is to describe a univer ...
GENTLY KILLING S–SPACES 1. Introduction and Notation In
GENTLY KILLING S–SPACES 1. Introduction and Notation In

... in which there are neither Q–sets nor locally compact, locally countable, hereditarily normal S–spaces. We provide such a model in this paper. In fact, in our model 2ℵ0 < 2ℵ1 (so in particular there are no Q–sets) and there are no locally compact, first countable S–spaces at all (hence no locally co ...
CLASSIFYING THE TYPES OF PRINCIPAL GROUPOID C
CLASSIFYING THE TYPES OF PRINCIPAL GROUPOID C

A study of remainders of topological groups
A study of remainders of topological groups

Free full version - topo.auburn.edu
Free full version - topo.auburn.edu

Grothendieck topology

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as l-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry.There is a natural way to associate a site to an ordinary topological space, and Grothendieck's theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namely sobriety, this is completely accurate—it is possible to recover a sober space from its associated site. However simple examples such as the indiscrete topological space show that not all topological spaces can be expressed using Grothendieck topologies. Conversely, there are Grothendieck topologies which do not come from topological spaces.