Ivancevic_Applied-Diff-Geom

8 Applied Differential Geometry: A Modern Introductionchanges to be diffeomorphisms, we get differentiable manifolds, or smoothmanifolds.We call two atlases compatible if the charts in the two atlases are allcompatible (or equivalently if the union of the two atlases is an atlas).Usually, we want to consider two compatible atlases as giving rise to thesame space. Formally, (as long as our concept of compatibility for chartshas certain simple properties), we can define an equivalence relation on theset of all atlases, calling two the same if they are compatible. In fact, theunion of all atlases compatible with a given atlas is itself an atlas, calleda complete (or maximal) atlas. Thus every atlas is contained in a uniquecomplete atlas.By definition, a smooth differentiable structure (or differential structure)on a manifold M is such a maximal atlas of charts, all related bysmooth coordinate changes on the overlaps.1.1.2 Topological ManifoldsA topological manifold is a manifold that is glued together from Euclideanspaces R n . Euclidean spaces are the simplest examples of topological manifolds.Thus, a topological manifold is a topological space that locally lookslike an Euclidean space. More precisely, a topological manifold is a topologicalspace 7 locally homeomorphic to a Euclidean space. This means that7 Topological spaces are structures that allow one to formalize concepts such as convergence,connectedness and continuity. They appear in virtually every branch of modernmathematics and are a central unifying notion. Technically, a topological space is a setX together with a collection T of subsets of X satisfying the following axioms:(1) The empty set and X are in T ;(2) The union of any collection of sets in T is also in T ; and(3) The intersection of any pair of sets in T is also in T .The collection T is a topology on X. The sets in T are the open sets, and theircomplements in X are the closed sets. The elements of X are called points. By induction,the intersection of any finite collection of open sets is open. Thus, the third Axiom can bereplaced by the equivalent one that the topology be closed under all finite intersectionsinstead of just pairwise intersections. This has the benefit that we need not explicitlyrequire that X be in T , since the empty intersection is (by convention) X. Similarly, wecan conclude that the empty set is in T by using Axiom 2. and taking a union over theempty collection. Nevertheless, it is conventional to include the first Axiom even whenit is redundant.A function between topological spaces is said to be continuous iff the inverse imageof every open set is open. This is an attempt to capture the intuition that there areno ‘breaks’ or ‘separations’ in the function. A homeomorphism is a bijection that iscontinuous and whose inverse is also continuous. Two spaces are said to be homeomor-