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What is Topology?
Sabino High School Math Club
Geillan Aly
University of Arizona
March 6, 2009
Math is Hard
• Mathematicians make math difficult:
 Formal language
Math is Hard
• Mathematicians make math difficult:
 Formal language
 Build on definitions and axioms
Solving Problems
• Express difficult concepts in terms of ideas
that are well understood
Solving Problems
• Express difficult concepts in terms of ideas
that are well understood
• Mathematics is mostly about determining
the “sameness” of two ideas
Sameness
• Algebra:
 Determine the sameness of two algebraic
structures.
Sameness
• Algebra:
 Determine the sameness of two algebraic
structures.
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Sameness
• Analysis:
 Given a function that cannot be calculated
easily, make an estimation in terms of
functions that can be calculated.
Sameness
• Analysis:
 Given a function that cannot be calculated
easily, make an estimation in terms of
functions that can be calculated.
Sameness
• Topology
 Determine the sameness of two geometric
objects
Sameness
• Topology
 Determine the sameness of two geometric
objects
• One can understand a difficult object if it is
related to a well understood subject.
Example
• The Poincaré Conjecture:
• Proven in 2005
 Every compact 3D simply connected manifold
without boundary is homeomorphic to a 3sphere.
Definitions
• What do we mean when we say “two
geometric objects are the same”?
Definitions
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Topology
Open Set
Closed Set
Continuity
Homeomorphic
Topology
• A Topology on a set X is a collection T of
subsets of X where:
 Ø and X are in T
Topology
• A Topology on a set X is a collection T of
subsets of X where:
 Ø and X are in T
 The union of elements in T are in T
Topology
• A Topology on a set X is a collection T of
subsets of X where:
 Ø and X are in T
 The union of elements in T are in T
 The intersection of any finite subcollection of
T is in T
Topology
• A Topology on a set X is a collection T of
subsets of X where:
 Ø and X are in T
 The union of elements in T are in T
 The intersection of any finite subcollection of
T is in T
• A set X where a topology has been
specified is a Topological Space.
Example
The three point set {red, yellow, blue} has 9 possible
topologies.
Topology
• Question:
The following examples are not topologies. Why?
Classifiying Sets
• A subset U of X is called Open if U is in T.
Classifiying Sets
• A subset U of X is called Open if U is in T.
• A subset V of X is called Closed if the
complement of V is in T.
Open and Closed Sets
Continuity
• A function f from one topological space X
to another Y is Continuous if f -1(U) is
open in X for every open set U in Y.
Continuity
• A function f from one topological space X
to another Y is Continuous if f -1(U) is
open in X for every open set U in Y.
Homeomorphism
• f : X Y is a homeomorphism if X and Y are
topological spaces and both f and f -1 are
continuous.
Homeomorphism
• f : X Y is a homeomorphism if X and Y are
topological spaces and both f and f -1 are
continuous.
• Two topological spaces are the “same” or
homeomorphic if there exists a homeomorphism
from one space to the other.
Homeomorphism
• f : X Y is a homeomorphism if X and Y are
topological spaces and both f and f -1 are
continuous.
• Two topological spaces are the “same” or
homeomorphic if there exists a homeomorphism
from one space to the other.
• It is easier to tell that two spaces are NOT
homeomorphic. Homeomoprhic spaces have
certain characteristics.
Homeomorphic
• Homeomorphic spaces can be visualized by
stretching, folding, and bending one space to
another. Think of topology as the ‘rubber’ subject.
Just don’t pinch, break or cut.
Homeomorphic
• Homeomorphic spaces can be visualized by
stretching, folding, and bending one space to
another. Think of topology as the ‘rubber’ subject.
Just don’t pinch, break or cut.
Homeomorphic Spaces?
Homeomorphic Spaces?
Homeomorphic Spaces?