TopologyUniformly ContinuousSequencesOpen and ClosedNeighborhoodHomeomorphismSmoothnessDiffeomorphismTopologyTopologyUniformly ContinuousUniformly ContinuousThe field of The field of topologytopology is about the geometrical study of is about the geometrical study of continuity.continuity.A map A map ff from a metric space ( from a metric space (XX, , dd) to a metric space ) to a metric space ((YY, , DD) is continuous if:) is continuous if:•Given Given 0, 0,   > 0, > 0,•If If dd((xx11, , xx22) < ) < , then , then dd(f((f(xx11), f(), f(xx22)) < )) < ..SequencesSequencesMap from positive integers to Map from positive integers to a set; s: a set; s: ZZ++ →→ XX•nnth point is th point is xxnn  ss((nn) ) Sequence {Sequence {xxnn} converges to } converges to yy•given given  > 0 there exists > 0 there exists mm•d(d(xxnn, , yy) < ) <  for for nn  mmLimit point Limit point pp in in YY  XX•There exists a sequence of There exists a sequence of points in points in YY converging to converging to ppy1XYynpOpen and ClosedOpen and ClosedA set A set YY  XX is is closedclosed if it if it contains all its limit points.contains all its limit points.The closure of The closure of YY, cl(, cl(YY) is the ) is the set of all limit points in set of all limit points in YY..BallBall of radius of radius rr centered at centered at xx•BB((xx, , rr) = {) = {yy: : yy  XX, , dd((xx, , yy) < ) < rr}}A set A set UU  XX is is openopen •For each For each xx  UU  rr•The ball The ball BB((xx, , rr) )  U U•Radius Radius rr may depend on may depend on xxClosed set includes its boundaryOpen set missing its boundaryNeighborhoodNeighborhoodThe interior of The interior of SS  YY•Int(Int(SS) = ) = YY – cl( – cl(YY - - SS))The neighborhood of a pointThe neighborhood of a point•xx  XX•NN  XX•xx  int( int(NN))YXNxInt(S)HomeomorphismHomeomorphismA function A function ff is continuous if is continuous if•XX and and YY are metric spaces are metric spaces•The function The function ff: : XX  YY•The function The function ff-1-1: : YY  XX open open VV  YY, , ff-1-1((VV) is open) is openA homeomorphism A homeomorphism ff•If If ff is continuous is continuous •and and ff is invertible is invertible•and and ff-1-1 is continuous is continuousfXYVf-1SmoothnessSmoothnessScalar field maps from a Scalar field maps from a space to the real numbers.space to the real numbers.•  = = ff((xx11, , xx22, , xx33))A constraint can reduce the A constraint can reduce the variables to a surface.variables to a surface.• FF((xx11, , xx22, , xx33) = 0) = 0•  = = ff((uu11, , uu22))Smooth fields are measured Smooth fields are measured by their differentiability.by their differentiability.•CCnn-smooth is -smooth is nn times times differentiabledifferentiable13212211cossinsincossinuaxuuaxuuax 211uuis not C2 smoothis C2 smoothDiffeomorphismDiffeomorphismA function A function  is class is class CCnn•XX  EEaa and and YY  EEbb•The function The function : : XX  YY•Open setsOpen sets U U  XX, , VV  YY•The function The function : : UU  VV•Partial derivatives of Partial derivatives of  are are continuous to order continuous to order nn•  and and  agree on agree on XXA function A function  is smoothis smooth•Class Class CCnn for all for all nnVUYX  is a is a CCnn-diffeomorphism-diffeomorphism•and and  is invertible is invertible•and and -1-1 is class is class