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... We have a short list of properties, but here are some of them. Proposition 2.1 Use the notation and objects we have defined above. 1. If X is the topological space consisting of only one point, then Hn (X) = 0 (the trivial group) if n > 0, and H0 (X) ∼ = Z. 2. If X is nonempty and path connected, th ...
... We have a short list of properties, but here are some of them. Proposition 2.1 Use the notation and objects we have defined above. 1. If X is the topological space consisting of only one point, then Hn (X) = 0 (the trivial group) if n > 0, and H0 (X) ∼ = Z. 2. If X is nonempty and path connected, th ...
Exercises on weak topologies and integrals
... G×V∗ →V∗ on the dual space and show that it is also continuous when V ∗ has the weak star topology. 4. Let V = C o (R) be the Frechét space of continuous functions on R, with seminorms given by suprema on compacta. (Since there is a countable cofinite set of these, the topology is Frechét.) Let G ...
... G×V∗ →V∗ on the dual space and show that it is also continuous when V ∗ has the weak star topology. 4. Let V = C o (R) be the Frechét space of continuous functions on R, with seminorms given by suprema on compacta. (Since there is a countable cofinite set of these, the topology is Frechét.) Let G ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.