MANIFOLDS AND CONNECTEDNESS Proposition 1. Let X be a
... Proposition 1. Let X be a topological manifold. Then X is locally connected. In other words, for every x ∈ X, x ∈ U , U open in X, there is a connected open set V so that x ∈ V ⊂ U . Remark. This is not true for all topological spaces Y . An example is Y = {0, 1, 1/2, 1/3, 1/4, . . . } with the subs ...
... Proposition 1. Let X be a topological manifold. Then X is locally connected. In other words, for every x ∈ X, x ∈ U , U open in X, there is a connected open set V so that x ∈ V ⊂ U . Remark. This is not true for all topological spaces Y . An example is Y = {0, 1, 1/2, 1/3, 1/4, . . . } with the subs ...
FA - 2
... in K, topologised by the norm kxk∞ = supn {|xn |}. Show that the subspace consisting of sequences that vanish at all but finitely many points is a convex balanced set which is not absorbing. (9) Let G be a compact abelian metric topological group (i.e., G is a metric space with an abelian group stru ...
... in K, topologised by the norm kxk∞ = supn {|xn |}. Show that the subspace consisting of sequences that vanish at all but finitely many points is a convex balanced set which is not absorbing. (9) Let G be a compact abelian metric topological group (i.e., G is a metric space with an abelian group stru ...
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... Examples The long line (without initial point) is homogeneous, but it is not bihomogeneous as its self-homeomorphisms are all order-preserving. This can be considered a pathological example, as most homogeneous topological spaces encountered in practice are also bihomogeneous. Every topological grou ...
... Examples The long line (without initial point) is homogeneous, but it is not bihomogeneous as its self-homeomorphisms are all order-preserving. This can be considered a pathological example, as most homogeneous topological spaces encountered in practice are also bihomogeneous. Every topological grou ...
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.