Topology

Topology

Senior Math PresentationNate Black

Bob Jones University 11/19/07

History

• Leonard Euler– Königsberg Bridge Problem

Königsberg Bridge Problem

J. J. O’Connor, A history of Topology

Königsberg Bridge Problem

A

B

C

D

Vertex Degree

A 3

B 5

C 3

D 3

A graph has a path traversing each edge exactlyonce if exactly two vertices have odd degree.

Königsberg Bridge Problem

A

B

C

D

Vertex Degree

A 3

B 4

C 3

D 2

A graph has a path traversing each edge exactlyonce if exactly two vertices have odd degree.

History

• Leonard Euler– Königsberg Bridge Problem

• August Möbius– Möbius Strip

Möbius Strip

• A sheet of paper has two sides, a front and a back, and one edge

• A möbius strip has one side and one edge

Möbius Strip

Plus Magazine ~ Imaging Maths – Inside the Klein Bottle

History

• Leonard Euler– Königsberg Bridge Problem

• August Möbius– Möbius Strip

• Felix Klein– Klein Bottle

Klein Bottle

• A sphere has an inside and an outside and no edges

• A klein bottle has only an outside and no edges

Klein Bottle

Plus Magazine ~ Imaging Maths – Inside the Klein Bottle

General Topology Overview

General Topology Overview

• Definition of a topological space• A topological space is a pair of objects,

, where is a non-empty set and is a collection of subsets of , such that the following four properties hold:– 1. – 2. – 3. If then– 4. If for each then

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General Topology Overview

• Terminology– is called the underlying set– is called the topology on– All the members of are called open sets

• Examples– with– Another topology on ,– The real line with open intervals, and in

general

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n

General Topology Overview

• Branches– Point-Set Topology

• Based on sets and subsets• Connectedness• Compactness

– Algebraic Topology• Derived from Combinatorial Topology• Models topological entities and relationships as

algebraic structures such as groups or a rings

General Topology Overview

• Definition of a topological subspace

• Let , and be topological spaces, then is said to be a subspace of

• The elements of are open sets by definition, if we let , then and these sets are said to be relatively open in

XY ,,, YX

YX

nOOO ,...,, 21YOO

11

nOOO ,...,, 21

Y

General Topology Overview

• Definition of a relatively open set

• Let , where is a subspace of and is a subset of . Then is said to be relatively open in if ,and is open in .

• Definition of a relatively closed set

• Let , where is a subspace of and is a subset of . Then is said to be relatively closed in if ,and is closed in .

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M X

Y X

YY

X

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General Topology Overview

• Definition of a neighborhood of a point

• Let , where is a topological space. Then a neighborhood of , denoted , is a subset of that contains an open set containing .

• Continuous function property

• A continuous function maps open/closed sets in into open/closed sets in

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a

X

X Y

Connectedness

Connectedness

• The general idea that all of the space touches. A point can freely be moved throughout the space to assume the location of any other point.

A

B

Connectedness

• General Connectedness– A space that cannot be broken up into several

disjoint yet open sets– Consider where

x1x2

A

B

Xx ,, 1 21, xxX

Connectedness

• General Connectedness– A space that cannot be broken up into several

disjoint yet open sets

• Path Connectedness– A space where any two points in the space

are connected by a path that lies entirely within the space

– This is different than a convex region where the path must be a straight line

Connectedness

• Simple Connectedness– A space that is free of “holes”– A space where every ball can be shrunk to a

point– A space where every path from a point A to a

point B can be deformed into any other path from the point A to the point B

General Connectedness

• Definition of general connectedness

• A topological space is said to be connected if , where is both open and closed, then or .

Xx

Xx

x X

General Connectedness

• Example of a disconnected set– with– Let– Then– This implies that is the complement of– Since is open then is closed– But is also open since it is an element of – Since is neither nor , is shown to be

disconnected

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4,3,2,1 BA

BAXBA ,B A

B

B

A

B X X

General Connectedness

• A subset of a topological space is said to be connected if , where is both relatively open and relatively closed, then .

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or , Aa

a

Path Connectedness

• Definition of a path in

• Let , , where is a continuous function, and let and . Then is called a path in and , the image of the interval, is a curve in that connects to .

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af 0

bf 1 f X 1,0f

X a b

X

0

1

a

b

f

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Path Connectedness

• Definition of a path connected space

• Let , where is a topological space. Then is said to be path connected if there is a path that connects to for all .

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x y

yx,

X

Path Connectedness

• Is every path connected space also generally connected?

• Let be a topological space that is path connected. Now suppose that is disconnected.

• Then is both open and closed, and or .

XX

AXA A X

Path Connectedness

• Let and . Since is path connected .

• Consider , clearly since .

• In addition, since .

• This set is then either open or closed but not both since is connected.

• Therefore, can be open or closed but not both.

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A

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Path Connected

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Path Connectedness

• This is a contradiction, so we conclude that every path connected space is also generally connected.

• Proof taken from Mendelson p. 135

Simple Connectedness

• Definition of a homotopy

• Let be paths in that connect to , where , then is said to be homotopic to if , where is continuous, such that the following hold true for .

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Simple Connectedness

(0,1) (1,1)

(1,0)

time

space

Mile marker 0 Mile marker 1

Time 0:

Mile marker 0 Mile marker 1

Time 1:

Simple Connectedness

(0,1) (1,1)

(1,0)

time

space1f

2f

Simple Connectedness

• The function is called the homotopy connecting to . and both belong to the same homotopy class.

• In a simply connected space any path between two points can be deformed into any other space.

• Consider the closed loops, ones in which the starting and ending points are the same. Then they must all be deformable into one another.

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Simple Connectedness

• One such closed path where we leave from a point A and return to it, is to never leave it.

• This path is called the constant path and is denoted by .Ae

Simple Connectedness

• Define a simply connected space

• Let be a topological space and . Then is said to be simply connected if for every there is only one homotopy class of closed paths. Since the constant path is guaranteed to be a closed path for , the homotopy class must be .

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x

Ae

Simply Connected

Exit

Entrance

Simply Connected

Compactness

Compactness

• Definition of a covering

• Let be a set, , and be an indexed subset of . Then the set is said to cover if .

• If only a finite number of sets are needed to cover , then is more specifically a finite covering.

XAX IBC

A C

A Cc

cA

A C

Compactness

• Definition of a compact space

• Let be a topological space, and let be a covering of . Then if for and is finite, then is said to be compact.

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X CDDC D X

Compactness

• Example of a space that is not compact

• Consider the real line

• The set of open intervals is clearly a covering of .

• Removing any one interval leaves an integer value uncovered.

• Therefore, no finite subcovering exists.

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Compactness

0 5-5

2 is no longer covered

Remove any interval

Compactness

• Define locally compact

• Let be a topological space, then is said to be locally compact if is compact.

• Note that every generally compact set is also locally compact since some subset of the finite coverings of the whole set will be a finite covering for some neighborhood of every in the space.

xNxNXx

X X

x

Compactness

• Is every closed subset of a compact space compact as well?

• Let be a closed subset of the compact space .

• If is an open covering of , then by adjoining the open set to the open covering of we obtain an open covering of .

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IU F FCO

J

V

X

Compactness

• Since is compact there is a finite subcovering of .

• However, each is either equal to a for some or equal to .

• If occurs among we may delete it to obtain a finite collection of the ’s that covers

• Proof taken from Mendelson 162-163

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21

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I O

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Applications

Applications

• Network Theory

• Knot Theory

• Genus categorization

Applications

• Genus = the number of holes in a surface– Formally: the largest number of

nonintersecting simple closed curves that can be drawn on the surface without separating it. (Mathworld)

5 holes

Matt Black

Applications

Matt Black

Applications

2006 Encyclopædia Britannica

Applications

• Definition of Fixed Point Theorem

• Let be a continuous function over the interval , such that and . Then .

f 1,0 00 f 11 f

zzfz 1,0

(1,1)(0,1)

(1,0)z

Applications

• is path connected, and since is continuous, is also path connected.

• If , then .

• Suppose never crosses the line then let be an interval containing some , where and .

1,0 f 1,0f

1,0v 1,0fvf

f xy ba,

z aaf bfb

Applications

• Clearly, there is a path that connects to since is a closed interval subset of and is therefore connected.

• Since is continuous we now shrink the width of the interval .

• When , the width will be zero and we can substitute in for and .

• This implies that and, since a path must connect the endpoints, .

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1,0f

f

bza z a b

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ba,

Applications

• Definition of a Conservative Force

• The work done in changing an object’s position is said to be path independent.

• One significant result of this type of a force is that the work done in moving an object along a closed path is 0.

Applications

• Let be a point in a space that is topologically simple, then is the homotopy class for all closed paths that include .

• The work done in not moving an object is obviously 0, and since all closed paths containing are in the same homotopy class, the work done on them is also 0.

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x

x

Applications

• We can prove it another way as well

• Let us define positive work to be that done on a path heading away from us, and negative work to be done when heading in the opposite direction, toward us.

• Then pick any point on the closed path as your starting point and some other point on the path as your ending point.

Applications

• Move along the straight line connecting those two points, and then move back along the same path.

• The total work done during the journey will be 0 since the magnitude of both trips is the same, but the signs are opposite.

• Each traversal of the path is homotopically equivalent to traversing one-half of the closed path.

Applications

+ work

– work

0 total work

Questions

Special Thanks

• Dr. Knisely for assistance in my paper direction and revision

• My brother, Matt Black, for making graphics I couldn’t find online

Bibliography• Königsberg city map

– http://www-history.mcs.st-and.ac.uk/HistTopics/Topology_in_mathematics.html

• Klein Bottle Images and rotating Möbius strip– http://plus.maths.org/issue26/features/mathart/index-gifd.html

• Möbius steps and genus diagrams and video– Matthew Black

• Coffee Cup Animation– http://www.britannica.com/eb/article-9108691

• Genus definition– http://mathworld.wolfram.com/Topology.html

• Mendelson Textbook– Mendelson, Bert. Introduction to Topology. NY: Dover, 1990.

• Munkers Textbook– Munkers, James R. Topology. New Delhi: Prentice Hall of India, 2000.

• All else, property of Nathanael Black