Chapter 5
... The locally compact Hausdorff case is nicer. First of all the condition on A ⊂ C(X) to separate the closed subsets of X (which may be difficult to prove!) can be reduced to a local condition. Theorem 5.19. Let X be a Hausdorff paracompact space and A ⊂ C(X) closed under locally finite sums and under ...
... The locally compact Hausdorff case is nicer. First of all the condition on A ⊂ C(X) to separate the closed subsets of X (which may be difficult to prove!) can be reduced to a local condition. Theorem 5.19. Let X be a Hausdorff paracompact space and A ⊂ C(X) closed under locally finite sums and under ...
Topology I
... 1. Let V be a vector space over ». A subset X of V is called convex if for any A and B in X, the segment AB = {tA + sB : s + t = 1} is a subset of X. 1a. Show that any subset X of V is contained in a smallest convex subset C(X) of V (called the convex hull of X). (5 pts.) 1b. Let Ai = (0, ..., 0, 1, ...
... 1. Let V be a vector space over ». A subset X of V is called convex if for any A and B in X, the segment AB = {tA + sB : s + t = 1} is a subset of X. 1a. Show that any subset X of V is contained in a smallest convex subset C(X) of V (called the convex hull of X). (5 pts.) 1b. Let Ai = (0, ..., 0, 1, ...
Lectures on Klein surfaces and their fundamental group.
... where π1 (X; x) is the topological fundamental group of X. The exact sequence above says that π1R ((X, σ); x) is in fact the orbifold fundamental group of X/σ, first defined in terms of orbifold paths by Thurston (see [Rat06] for a presentation of the general theory of orbifolds and Thurston’s defin ...
... where π1 (X; x) is the topological fundamental group of X. The exact sequence above says that π1R ((X, σ); x) is in fact the orbifold fundamental group of X/σ, first defined in terms of orbifold paths by Thurston (see [Rat06] for a presentation of the general theory of orbifolds and Thurston’s defin ...
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.