Neighbourhood Systems - PMF-a
... Definition 3.3. Let (X, L, N) and (Y, T, M) be two neighbourhood systems. A continuous map from (X, L, N) to (Y, T, M) is a pair ( f, f¯) where f : X → Y is a function, f¯ : T → L is a frame homomorphism and for every x ∈ X, u ∈ M( f (x)) if and only if f¯(u) ∈ N(x). Example 3.4. Let f : R → R be a ...
... Definition 3.3. Let (X, L, N) and (Y, T, M) be two neighbourhood systems. A continuous map from (X, L, N) to (Y, T, M) is a pair ( f, f¯) where f : X → Y is a function, f¯ : T → L is a frame homomorphism and for every x ∈ X, u ∈ M( f (x)) if and only if f¯(u) ∈ N(x). Example 3.4. Let f : R → R be a ...
A VERY BRIEF INTRODUCTION TO ERGODIC THEORY 1
... This approach allowed Furstenberg [2] to give a dynamical proof for the following theorem Theorem 1.14 (Weyl). Let p(x) = ad xd + · · · + a1 x + a0 be a polynomial with at least one irrational nonconstant coefficient. Then {p(n)}∞ n=0 is equi-distributed on T. 1.4. Additional remarks and exercises. ...
... This approach allowed Furstenberg [2] to give a dynamical proof for the following theorem Theorem 1.14 (Weyl). Let p(x) = ad xd + · · · + a1 x + a0 be a polynomial with at least one irrational nonconstant coefficient. Then {p(n)}∞ n=0 is equi-distributed on T. 1.4. Additional remarks and exercises. ...
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.