Section 11.5. Compact Topological Spaces
... Note. We now present results similar to results seen in the metric space setting, but slightly different because they require specific properties of a topological space (such as Hausdorff). ...
... Note. We now present results similar to results seen in the metric space setting, but slightly different because they require specific properties of a topological space (such as Hausdorff). ...
Solutions to selected exercises
... c. For the ‘only if’ or =⇒ direction, let x ∈ A and let N be a neighbourhood of x. Then there exists an open set U such that x ∈ U ⊂ N . If U ∩ A = ∅, then X \ U is a closed set containing A, therefore x ∈ A ⊂ X \ U . But this contradicts the fact that x ∈ U . Hence we must have U ∩ A 6= ∅, and so N ...
... c. For the ‘only if’ or =⇒ direction, let x ∈ A and let N be a neighbourhood of x. Then there exists an open set U such that x ∈ U ⊂ N . If U ∩ A = ∅, then X \ U is a closed set containing A, therefore x ∈ A ⊂ X \ U . But this contradicts the fact that x ∈ U . Hence we must have U ∩ A 6= ∅, and so N ...
PROOF. Let a = ∫X f dµ/µ(X). By convexity the graph of g lies
... D EFINITION A.6.23. A topological manifold is a Hausdorff space X with a countable base for the topology such that every point is contained in an open set homeomorphic to a ball in Rn . A pair (U , h) of such a neighborhood and a homeomorphism h : U → B ⊂ Rn is called a chart or a system of local co ...
... D EFINITION A.6.23. A topological manifold is a Hausdorff space X with a countable base for the topology such that every point is contained in an open set homeomorphic to a ball in Rn . A pair (U , h) of such a neighborhood and a homeomorphism h : U → B ⊂ Rn is called a chart or a system of local co ...
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.