Proper Maps and Universally Closed Maps
Proper Maps and Universally Closed Maps

equivariant homotopy and cohomology theory
equivariant homotopy and cohomology theory

... There is a great deal of literature on this subject. The original construction of the nonequivariant stable homotopy category was due to Mike Boardman. One must make a sharp distinction between the stable homotopy category, which is xed and unique up to equivalence, and any particular point-set lev ...
Sβ−COMPACTNESS IN L-TOPOLOGICAL SPACES
Sβ−COMPACTNESS IN L-TOPOLOGICAL SPACES

topology - DDE, MDU, Rohtak
topology - DDE, MDU, Rohtak

An addendum to Kelley`s "General Topology" written by myself that
An addendum to Kelley`s "General Topology" written by myself that

Morphisms in Logic, Topology, and Formal Concept Analysis
Morphisms in Logic, Topology, and Formal Concept Analysis

Real Analysis
Real Analysis

Some Stronger Forms of gb –continuous Functions
Some Stronger Forms of gb –continuous Functions

Foldings and Deformation Retractions of Hypercylinder
Foldings and Deformation Retractions of Hypercylinder

TOPOLOGY 004C Contents 1. Introduction 1 2. Basic notions 2 3
TOPOLOGY 004C Contents 1. Introduction 1 2. Basic notions 2 3

APPLICATIONS OF THE TARSKI–KANTOROVITCH FIXED
APPLICATIONS OF THE TARSKI–KANTOROVITCH FIXED

NOTES WEEK 13 DAY 1 Unassigned exercise: Let X and Y be
NOTES WEEK 13 DAY 1 Unassigned exercise: Let X and Y be

More on Generalized Homeomorphisms in Topological Spaces
More on Generalized Homeomorphisms in Topological Spaces

On πgr-Closed Sets in Ideal Topological Spaces
On πgr-Closed Sets in Ideal Topological Spaces

Elementary Topology Problem Textbook O. Ya. Viro, OA
Elementary Topology Problem Textbook O. Ya. Viro, OA

Convexity of Hamiltonian Manifolds
Convexity of Hamiltonian Manifolds

... shows one direction, the other is trivial. Putting u = v in the definition, we see that for a convex manifold M the fibers of ψ are connected. Moreover, for every u, v ∈ ψ(M ) the intersection uv ∩ ψ(M ) has to be connected, i.e., ψ(M ) is convex. These two properties are not enough to imply convexi ...
THE HOMOMORPHISMS OF TOPOLOGICAL GROUPOIDS 1
THE HOMOMORPHISMS OF TOPOLOGICAL GROUPOIDS 1

PDF file
PDF file

Compact-like properties in hyperspaces
Compact-like properties in hyperspaces

Near topology and nearly continuous functions
Near topology and nearly continuous functions

... This Dissertation is brought to you for free and open access by Digital Repository @ Iowa State University. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more information, please ...
closed sets via generalized topologies
closed sets via generalized topologies

... basic operators on generalized topological spaces. It is observed that a large number of papers are devoted to the study of generalized open sets like open sets of a topological space, containing the class of open sets and possessing properties more or less similar to those of open sets. We recall s ...
Complex cobordism of Hilbert manifolds with some applications to
Complex cobordism of Hilbert manifolds with some applications to

Extremally Disconnectedness in Ideal Bitopological Spaces
Extremally Disconnectedness in Ideal Bitopological Spaces

On the Decomposition of δ -β-I-open Set and Continuity in the Ideal
On the Decomposition of δ -β-I-open Set and Continuity in the Ideal

Compact factors in finally compact products of topological spaces
Compact factors in finally compact products of topological spaces

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.