NOTES ON GENERAL TOPOLOGY 1. The notion of a topological
... Part of the rigorization of analysis in the 19th century was the realization that notions like convergence of sequences and continuity of functions (e.g. f : Rn → Rm ) were most naturally formulated by paying close attention to the mapping properties between subsets U of the domain and codomain with ...
... Part of the rigorization of analysis in the 19th century was the realization that notions like convergence of sequences and continuity of functions (e.g. f : Rn → Rm ) were most naturally formulated by paying close attention to the mapping properties between subsets U of the domain and codomain with ...
Normality on Topological Groups - Matemáticas UCM
... discrete in X. In order to see that Y is discrete, it is enough to define a continuous character in G which takes the value −1 in yα and 1 in yβ for every β ∈ A \ {α}. Consider first ϕ : Z → T, the character on Z such that ϕ(n) = eπni , ∀n ∈ Z. It gives rise to characters on ZR , just composing with ...
... discrete in X. In order to see that Y is discrete, it is enough to define a continuous character in G which takes the value −1 in yα and 1 in yβ for every β ∈ A \ {α}. Consider first ϕ : Z → T, the character on Z such that ϕ(n) = eπni , ∀n ∈ Z. It gives rise to characters on ZR , just composing with ...
Class 3 - Stanford Mathematics
... We now briefly introduce a few notions about abelian categories. We will soon define some new categories (certain sheaves) that will have familiar-looking behavior, reminiscent of that of modules over a ring. The notions of kernels, cokernels, images, and more will make sense, and they will behave ...
... We now briefly introduce a few notions about abelian categories. We will soon define some new categories (certain sheaves) that will have familiar-looking behavior, reminiscent of that of modules over a ring. The notions of kernels, cokernels, images, and more will make sense, and they will behave ...
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a ""clockwise"" direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a ""clockwise"" orientation for all loops in the manifold. Equivalently, a surface is orientable if a two-dimensional figure such as 20px in the space cannot be moved (continuously) around the space and back to where it started so that it looks like its own mirror image 20px.The notion of orientability can be generalised to higher-dimensional manifolds as well. A manifold is orientable if it has a consistent choice of orientation, and a connected orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms. An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.