Transcript An Introduction to Topology: Connectedness and

Topology
Senior Math Presentation
Nate 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
Vertex
Degree
A
3
B
5
C
3
D
3
C
B
D
A
A graph has a path traversing each edge exactly
once if exactly two vertices have odd degree.
Königsberg Bridge Problem
Vertex
Degree
A
3
B
4
C
3
D
2
C
B
D
A
A graph has a path traversing each edge exactly
once 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,
 X ,  , where X is a non-empty set and  is
a collection of subsets of X , such that the
following four properties hold:
– 1. X  
– 2.   
– 3. If O1 , O2 ,...,On   then O1  O2  ... On  
– 4. If for each   I, O   then I O  
General Topology Overview
• Terminology
– X is called the underlying set
–  is called the topology on X
– All the members of  are called open sets
• Examples
– X  x1 , x2 , x3 , x4 , x5  with   , x1 , x2 , x1 , x2 , X 
– Another topology on X ,   , x1 , X 
– The real line with open intervals, and in
general  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 Y  X , and  X , , Y ,  be topological
spaces, then Y is said to be a subspace of
X
• The elements of   O1 , O2 ,...,On  are open

O
sets by definition, if we let  O  Y , then
 

  O , O ,...,O  and these sets are said to be
relatively open in Y
1
1
2
n
1
General Topology Overview
• Definition of a relatively open set
• Let , A  Y  X where Y is a subspace of X
and A is a subset of Y . Then A is said to
be relatively open in Y if ,  M  X  M  Y  A
and M is open in X.
• Definition of a relatively closed set
• Let , B  Y  X where Y is a subspace of X
and B is a subset of Y . Then B is said to
be relatively closed in Y if ,  N  X  N  Y  B
and N is closed in X.
General Topology Overview
• Definition of a neighborhood of a point
• Let a  X , where X is a topological space.
Then a neighborhood of a , denoted N (a ) ,
is a subset of X that contains an open set
containing a .
• Continuous function property
• A continuous function maps open/closed
sets in X into open/closed sets in 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.
B
A
Connectedness
• General Connectedness
– A space that cannot be broken up into several
disjoint yet open sets
– Consider X  x1 , x2  where   , x1, X 
x2
x1
B
A
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 X is said to be
connected if x  X, where x is both open
and closed, then x   or X .
General Connectedness
• Example of a disconnected set
– X  1,2,3,4 with   , 1,2, 3,4, 1,2,3,4
– Let A  1,2, B  3,4
– Then A  B  X , A  B  
– This implies that B is the complement of A
– Since A is open then B is closed
– But B is also open since it is an element of 
– Since B is neither X nor  , X is shown to be
disconnected
General Connectedness
• A subset A of a topological space X is said
to be connected if a  A, where a is both
relatively open and relatively closed, then
a  A , or  .
Path Connectedness
• Definition of a path in X
• Let a, b  X , f : 0,1  X , where f is a
continuous function, and let f 0  a and
f 1  b . Then f is called a path in X and
f 0,1 , the image of the interval, is a curve
in X that connects a to b .
0
f
f
1
a
b
Path Connectedness
• Definition of a path connected space
• Let x, y  X, where X is a topological space.
Then X is said to be path connected if
there is a path that connects x to y for all
x, y.
Path Connectedness
• Is every path connected space also
generally connected?
• Let X be a topological space that is path
connected. Now suppose that X is
disconnected.
• Then  A  X  A is both open and closed,
and A   or X .
Path Connectedness
• Let a  A and b  C  A. Since X is path
connected  f : 0,1  X  f 0  a, f 1  b .
• Consider B  t  f t   A , clearly B  
since f 0  a  A.
• In addition, B  0,1 since f 1  b  A .
• This set B is then either open or closed but
not both since 0,1 is connected.
• Therefore, A can be open or closed but not
both.
Path Connected
X
f
0
a
B
A
f
1
f
b
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 f1 , f 2 be paths in X that connect x to y ,
where x, y  X , then f1 is said to be
homotopic to f 2 if  H : 0,12  X, where H is
continuous, such that the following hold
true for 0  x, t  1 .
H 0, t   x
H 1, t   y
H s ,0   f 1 s 
H s,1  f 2 s 
Simple Connectedness
time
Time 0:
(0,1)
(1,1)
Mile marker 0
Mile marker 1
Mile marker 0
Mile marker 1
space
(1,0)
Time 1:
Simple Connectedness
time
(0,1)
f2
(1,1)
f1
space
(1,0)
Simple Connectedness
• The function H is called the homotopy
connecting f1 to f 2. f1 and f 2 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.
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 e A .
Simple Connectedness
• Define a simply connected space
• Let X be a topological space and x  X .
Then X is said to be simply connected if for
every x there is only one homotopy class
of closed paths. Since the constant path is
guaranteed to be a closed path for x , the
homotopy class must be eA  .
Simply Connected
Entrance
Exit
Simply Connected
Compactness
Compactness
• Definition of a covering
• Let X be a set, A  X , and C  B I be an
indexed subset of A . Then the set C is
said to cover A if A   c .
cC
• If only a finite number of sets are needed
to cover A , then C is more specifically a
finite covering.
Compactness
• Definition of a compact space
• Let X be a topological space, and let C  B I
be a covering of X . Then if for  C  D  D  C
and D is finite, then X is said to be
compact.
Compactness
• Example of a space that is not compact
• Consider the real line
• The set of open intervals n, n  2 n  Z is
clearly a covering of R .
• Removing any one interval leaves an
integer value uncovered.
• Therefore, no finite subcovering exists.
Compactness
Remove any interval
-5
0
5
2 is no longer covered
Compactness
• Define locally compact
• Let X be a topological space, then X is
said to be locally compact if  x  X  N x  N x
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 x in the space.
Compactness
• Is every closed subset of a compact space
compact as well?
• Let F be a closed subset of the compact
space X .
• If U I is an open covering of F , then by
adjoining the open set O  CF  to the open
covering of V J we obtain an open
covering of X .
Compactness
• Since X is compact there is a finite
subcovering V ,V ,...,V of X .
• However, each V is either equal to a U 
for some   I or equal to O .
• If O occurs among V ,V ,...,V we may
delete it to obtain a finite collection of the
U  ’s that covers F  C O
• Proof taken from Mendelson 162-163
1
2
m
i
1
2
m
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 f be a continuous function over the
interval 0,1, such that f 0  0 and f 1  1 .
Then  z  0,1  f z   z.
(1,1)
(0,1)
z
(1,0)
Applications
• 0,1 is path connected, and since f is
continuous, f 0,1 is also path connected.
• If v  0,1 , then f v   f 0,1 .
• Suppose f never crosses the line y  x
then let a, b be an interval containing some
z , where f a   a and b  f b.
Applications
• Clearly, there is a path that connects f a  to
f b  since f a, b is a closed interval subset
of f 0,1 and is therefore connected.
• Since f is continuous we now shrink the
width of the interval a, b.
• When a  z  b , the width will be zero and
we can substitute z in for a and b .
• This implies that f z   z  f z  and, since a
path must connect the endpoints, f z   z .
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 x be a point in a space that is
topologically simple, then x  X , ex  is the
homotopy class for all closed paths that
include x.
• The work done in not moving an object is
obviously 0, and since all closed paths
containing x are in the same homotopy
class, the work done on them is also 0.
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
0 total work
+ work
– 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