Topological Vector Spaces III: Finite Dimensional Spaces
... The continuity of (5) is immediate from Proposition 1. To prove the continuity of (6) we notice that, by the definition of the product topology (see TVS II), all we need to do is prove the continuity of the coordinate maps πi : (Kn+1 , T) 3 (α1 , . . . , αn+1 ) 7−→ αi ∈ K, i = 1, 2, . . . , n + 1. ...
... The continuity of (5) is immediate from Proposition 1. To prove the continuity of (6) we notice that, by the definition of the product topology (see TVS II), all we need to do is prove the continuity of the coordinate maps πi : (Kn+1 , T) 3 (α1 , . . . , αn+1 ) 7−→ αi ∈ K, i = 1, 2, . . . , n + 1. ...
The inverse map of a continuous bijective map might not be
... If X�is not assumed to be compact, then for a bijective map f : X → Y , f being continuous cannot ensure that f −1 is also continuous. One typical example is like this: Let X = [0, 1) and let Y = S 1 ⊂ C. Define f : X → Y, x 7→ e2πix . One can immediately check that f is continuous, f is bijective, b ...
... If X�is not assumed to be compact, then for a bijective map f : X → Y , f being continuous cannot ensure that f −1 is also continuous. One typical example is like this: Let X = [0, 1) and let Y = S 1 ⊂ C. Define f : X → Y, x 7→ e2πix . One can immediately check that f is continuous, f is bijective, b ...
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.