What is Topology?

1. What is Topology? Sabino High School Math Club Geillan Aly University of Arizona March 6, 2009

2. Math is Hard • Mathematicians make math difficult: • Formal language

3. Math is Hard • Mathematicians make math difficult: • Formal language • Build on definitions and axioms

4. Solving Problems • Express difficult concepts in terms of ideas that are well understood

5. Solving Problems • Express difficult concepts in terms of ideas that are well understood • Mathematics is mostly about determining the “sameness” of two ideas

6. Sameness • Algebra: • Determine the sameness of two algebraic structures.

7. Sameness • Algebra: • Determine the sameness of two algebraic structures.

8. Sameness • Analysis: • Given a function that cannot be calculated easily, make an estimation in terms of functions that can be calculated.

9. Sameness • Analysis: • Given a function that cannot be calculated easily, make an estimation in terms of functions that can be calculated.

10. Sameness • Topology • Determine the sameness of two geometric objects

11. Sameness • Topology • Determine the sameness of two geometric objects • One can understand a difficult object if it is related to a well understood subject.

12. Example • The Poincaré Conjecture: • Proven in 2005 • Every compact 3D simply connected manifold without boundary is homeomorphic to a 3-sphere.

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

14. Definitions • Topology • Open Set • Closed Set • Continuity • Homeomorphic

15. Topology • A Topology on a set X is a collection T of subsets of X where: • Ø and X are in T

16. 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

17. 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

18. 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.

19. Example The three point set {red, yellow, blue} has 9 possible topologies.

20. Topology • Question: The following examples are not topologies. Why?

21. Classifiying Sets • A subset U of X is called Open if U is in T.

22. 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.

23. Open and Closed Sets

24. 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.

25. 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.

26. Homeomorphism • f : X Y is a homeomorphism if X and Y are topological spaces and both f and f -1 are continuous.

27. 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.

28. 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.

29. 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.

30. 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.

31. Homeomorphic Spaces?

32. Homeomorphic Spaces?

33. Homeomorphic Spaces?