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Chapter 5 Hyperspaces
Chapter 5 Hyperspaces

Locally ringed spaces and manifolds
Locally ringed spaces and manifolds

Internal Hom-Objects in the Category of Topological Spaces
Internal Hom-Objects in the Category of Topological Spaces

... For ordinary sets A and B, denote by Hom(A, B) the set of functions from A to B. Given three sets X, Y, T and a map g : T × X → Y we can construct a second map ḡ : T → Hom(X, Y ) called the transpose of g, defined by ḡ(t)(x) = g(t, x). Every such map g induces a unique transpose in this way. Conve ...
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Shortest paths and geodesics

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Locally Convex Vector Spaces III: The Metric Point of View

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The Hilbert–Smith conjecture for three-manifolds

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... characterizing monotone normality, the role of points will now be played by the closure of singletons (the minimal closed sets in a non T1 -space). This idea is as simple as effective. It is also used to provide an extension property of lattice-valued functions for monotonically normal spaces. This ...
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Topological Spaces. - Dartmouth Math Home
Topological Spaces. - Dartmouth Math Home

Draft version F ebruary 5, 2015
Draft version F ebruary 5, 2015

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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.
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