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

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 *ABCD AABCD BBCDA CCDAB DDABC Algebra:  Determine the sameness of two algebraic structures.

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 3- sphere.

Definitions What do we mean when we say “two geometric objects are the same”?

Definitions 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?