Ivancevic_Applied-Diff-Geom

Introduction 9every point has a neighborhood for which there exists a homeomorphism(a bijective continuous function whose inverse is also continuous) mappingthat neighborhood to R n . These homeomorphisms are the charts of themanifold.Usually additional technical assumptions on the topological space aremade to exclude pathological cases. It is customary to require that thespace be Hausdorff and second countable.The dimension of the manifold at a certain point is the dimension of theEuclidean space charts at that point map to (number n in the definition).All points in a connected manifold have the same dimension.In topology and related branches of mathematics, a connected space isa topological space which cannot be written as the disjoint union of two ormore nonempty spaces. Connectedness is one of the principal topologicalproperties that is used to distinguish topological spaces. A stronger notionis that of a path–connected space, which is a space where any two pointscan be joined by a path. 8phic if there exists a homeomorphism between them. From the standpoint of topology,homeomorphic spaces are essentially identical.The category of topological spaces, Top, with topological spaces as objects and continuousfunctions as morphisms is one of the fundamental categories in mathematics. Theattempt to classify the objects of this category (up to homeomorphism) by invariants hasmotivated and generated entire areas of research, such as homotopy theory, homologytheory, and K–theory.8 Formally, for a topological space X the following conditions are equivalent:(1) X is connected.(2) X cannot be divided into two disjoint nonempty closed sets (this follows since thecomplement of an open set is closed).(3) The only sets which are both open and closed (open sets) are X and the empty set.(4) The only sets with empty boundary are X and the empty set.(5) X cannot be written as the union of two nonempty separated sets.The maximal nonempty connected subsets of any topological space are called theconnected components of the space. The components form a partition of the space(that is, they are disjoint and their union is the whole space). Every component is aclosed subset of the original space. The components in general need not be open: thecomponents of the rational numbers, for instance, are the one–point sets. A space inwhich all components are one–point sets is called totally disconnected.The space X is said to be path–connected iff for any two points x, y ∈ X there existsa continuous function f : [0, 1] → X, from the unit interval [0, 1] to X, with f(0) = xand f(1) = y (this function is called a path from x to y). Every path–connected spaceis connected.