Homework #4
... Work all these problems and talk to me if you have any questions on them, but carefully write up and turn in only problems 1, 2, 5, 7, 10 and 12. Due: November 8 1) Show a compact metric space is second countable and separable. Hint: Think totally bounded. This actually follows immediately from Prob ...
... Work all these problems and talk to me if you have any questions on them, but carefully write up and turn in only problems 1, 2, 5, 7, 10 and 12. Due: November 8 1) Show a compact metric space is second countable and separable. Hint: Think totally bounded. This actually follows immediately from Prob ...
(The Topology of Metric Spaces) (pdf form)
... If (S, d) is a metric space, we let T = TS be the set of open sets of the metric space. The set T is a collection of subsets of S that has the following properties: S (O1) If Ui ∈ T for i ∈ I, then i∈I Ui ∈ T ; (O2) If U, V ∈ T , then U ∩ V ∈ T ; (O3) ∅, S ∈ T . A collection T of subsets of a set S ...
... If (S, d) is a metric space, we let T = TS be the set of open sets of the metric space. The set T is a collection of subsets of S that has the following properties: S (O1) If Ui ∈ T for i ∈ I, then i∈I Ui ∈ T ; (O2) If U, V ∈ T , then U ∩ V ∈ T ; (O3) ∅, S ∈ T . A collection T of subsets of a set S ...
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.