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Dive into the realm of topology to uncover the "sameness" of geometric objects and the essence of homeomorphism. Discover how mathematicians simplify complex concepts and relate them to well-understood ideas. Explore the intricacies of open sets, closed sets, continuity, and homeomorphic spaces. Unravel the puzzle of topological spaces and the art of visualizing different shapes through stretching and folding.
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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.
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.
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.
