TopologyTopology

Uniformly 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 pp

y1

X

Y

yn

p

Open 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 xx

Closed set includes its boundary

Open set missing its boundary

NeighborhoodNeighborhood 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))

Y

XN x

Int(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 continuous

f

X

Y

Vf-1

SmoothnessSmoothness 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

differentiabledifferentiable

13

212

211

cossinsincossin

uaxuuaxuuax

21

1

u

u

is not C2 smooth

is C2 smooth

DiffeomorphismDiffeomorphism 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

U

Y

X

is a is a CCnn-diffeomorphism-diffeomorphism• and and is invertible is invertible• and and -1-1 is class is class CCnn

next