THE WEAK HOMOTOPY EQUIVALENCE OF Sn AND A SPACE
THE WEAK HOMOTOPY EQUIVALENCE OF Sn AND A SPACE

F A S C I C U L I M A T H E M A T I C I
F A S C I C U L I M A T H E M A T I C I

On Relative Preclosedness of Strongly Compact (Countably p
On Relative Preclosedness of Strongly Compact (Countably p

A generalization of realcompact spaces
A generalization of realcompact spaces

On Regular b-Open Sets in Topological Spaces
On Regular b-Open Sets in Topological Spaces

IV.2 Basic topological properties
IV.2 Basic topological properties

K - CIS @ UPenn
K - CIS @ UPenn

Some forms of the closed graph theorem
Some forms of the closed graph theorem

on topological chaos
on topological chaos

... Lebesgue measure. The definition used later by other authors is the one above, that extends to all metric spaces and does not contain probabilistic assumption. On the unit interval sensitivity ⇒ positive topological entropy ⇒ Li-Yorke chaos Transitivity We say that f is topologically transitive if t ...
Omega open sets in generalized topological spaces
Omega open sets in generalized topological spaces

Full
Full

Topological Vector Spaces and Continuous Linear Functionals
Topological Vector Spaces and Continuous Linear Functionals

Lecture Notes (unique pdf file)
Lecture Notes (unique pdf file)

What to remember about metric spaces
What to remember about metric spaces

Generically there is but one self homeomorphism of the Cantor set
Generically there is but one self homeomorphism of the Cantor set

Some Generalized Forms of Compactness
Some Generalized Forms of Compactness

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

Formal Connected Basic Pairs
Formal Connected Basic Pairs

... the basic neighbourhoods immediately connected to S in Y are the basic neighbourhoods having at least one point in Y (in S). Similarly, one proves that noccY X = extS ∩ Y , that is, the points of X which are “glued” to some other point via the basic neighbourhoods of Y are exactly the points of Y ly ...
Free full version - topo.auburn.edu
Free full version - topo.auburn.edu

Topology Proceedings - topo.auburn.edu
Topology Proceedings - topo.auburn.edu

On the Level Spaces of Fuzzy Topological Spaces
On the Level Spaces of Fuzzy Topological Spaces

Manifolds of smooth maps
Manifolds of smooth maps

What to remember about metric spaces KC Border CALIFORNIA INSTITUTE OF TECHNOLOGY
What to remember about metric spaces KC Border CALIFORNIA INSTITUTE OF TECHNOLOGY

... metrics generating the same topology are equivalent. The Euclidean, ℓ1 , and sup metrics on Rm are equivalent metrics for the topology of Rm . A property of a metric space that can be expressed in terms of open sets without mentioning a specific metric is called a topological property. It is possibl ...
RIEMANN SURFACES 2. Week 2. Basic definitions 2.1. Smooth
RIEMANN SURFACES 2. Week 2. Basic definitions 2.1. Smooth

... The meaning of the definitions is as follows. A chart f : U a coordinate system on U (a local coordinate system). A pair of compartible charts defines two coordinate systems on the intersection. Compatibility means that the passage from one coordinate system to the other is smooth. 2.1.1. Definition ...


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.