Martin Flashman

Department of Mathematics

Humboldt State University 

"What Is Topology and Why Is It Useful? "

Thursday, March 7, 2002   SH 128
Abstract: In trying to answer the questions posed by the title, Professor Flashman will present an introduction to some key topological concepts and examples of how a topological view can influence understanding of geometric and analytic results.  

 
  • First Responses to "what is it?":
  • Topology is
  • rubber sheet geometry ... the study of surfaces allowing strecthing and shrinking and some cutting and pasting.
  • Examples of "rubber sheet geometry."
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  • A generalized approach to geometric results without using measurement.
  • A generalized approach to analytic results based on the structure of "open" sets.
  • A collection of  problems that arise from using "topological" methods.
  • The study of the logical consequences related to the structural relations of "open sets" and "continuous functions."
  • Professor Yoon's Enquirer Article from the Encyclopedia.
  • Encyclopedia Responses.
  • Mathematical Atlas.
  • Many On-line resources.
  • Topology without Tears by Sidney A Morris
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  • First Responses to "why is it useful?"
  • It presents alternative foundations for mathematical results.
  • It provides alternative concepts and tools for understanding and solving  problems.
  • "Finding Topology in the Factory: Configuration Spaces" by Aaron Abrams and Robert Ghrist, American Mathematical Monthly, vol. 109, no. 2 (2002), pp. 140-150.
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  • Topological approaches to other studies.
  • Geometry
  • Graph theory
  • Real and Complex analysis
  • Differential Equations
  • Linear Algebra
  • Logic.
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  • Different types of topological studies.
  • Point Set Topology uses set theory. [It provides foundational concepts for geometry and analysis.]

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  • Geometric or Piecewise Linear (PL) Topology uses linear-geometric tools. Polygons and polyhedra are some examples of PL topology objects.

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  • Differential Topology uses the derivative and calculus tools. Surfaces and manifolds that are characterized by differentiable functions are the primary objects of differential topology.

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  • Algebraic Topology uses combinatorial and algebraic tools. Counting and the assignment of related algebraic objects, especially groups and algebras, to topological spaces with sufficiently nice structure are characteristics of algebraic topology.
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    		Objects Maps
    Geometry Points, Lines Isometries, Similarities,  
    Affine transformations, Projective Transformations
    Point set topology Sets, Open Sets Continuous functions
    Piecewise Linear (PL) Topology Simplicial Complexes, Cells PL functions 
    Differential Topology Differentaible Manifolds, Charts Differentiable, Analytic functions
    Algebraic Topology CW Complex, Cells Continuous functions
     
  • Examples of elementary applications:

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  • Calculus  (Use WinPlot for examples. )
  • IVT : An application of connectedness as a "topological" property.
  • EVT : An application of compactness as a "topological" property.
  • Fixed Point Theorem on [0,1] and Dn

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  • Geometry
  • Classification of Platonic Solids (Euclid Book 13 Prop.18):
  • Application of the "Euler" formula.(Use WinGeometry)
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  • Planar Graph Theory:
  • Konigsburg Bridges
  • Utility Problem
  • Color Problems
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  • Orientability of geometries:
  • Euclidean Plane: Yes
  • The Moebius Band: No.
  • Projective Plane: No.

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  • Topology
  • Classification of Surfaces.
  • The sphere
  • The Torus
  • Moebius Band
  • Projective Plane
  • Klein Bottle I and Klein Bottle II

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  • Jordan Curve Theorem:

    • A simple closed curve in the plane separates the plane into two disjoint open sets.
     
  • Decomposition Theory:
  • Example: Any compact surface without boundary is homeomorphic to a polyhedral surface made up entirely of triangles.

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  • Finite Topologies.
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  • What is a topological space? [Axiomatic characterization]:

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    A topological space is a  set X  together with a family of subsets T={O} called open sets with the properties:
     

  • The empty set and X are members of T.
  • The intersection of any pair of members of T is also a member of T.
  • The union of any family of members of T is also a member of T.

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  • If X and Y are topological spaces, what is a continuous function f: X ® Y?

    • Definition: A function f is continuous if the preimage of any open set of Y is an open set of X.
     
  • What does it mean to say two topological spaces are topologically equivalent (homeomorphic)?

    • Definition: X and Y are topologically equivalent if there are continuous functions f :X®Y and g:Y®X so that fg=id and gf=id.
    (f and g are inverse functions, f and g are 1:1 and onto.)
     
  • What does it mean to say A is a topological subspace of X?

    • Definition: A is a topological subspace of X if A is a topological space and the inclusion function i:A->X is a continuous function.
     
     
  • What does it mean to say a property of a topological space is inherited?

    • A property P is inherited if X has property P and if A is a subspace of X then A has property P.

     
  • Topological Properties: 
  • What does it mean to say that X is connected?

    • X is not connected if there is a continuous function onto the set {0,1} with the discrete topology... all subsets are open.
    I.e., There are disjoint open sets A and B of X  with X = A È B.
  • What does it mean to say that X is path connected?

    • X is path connected if for any two points a and b of X there is a continuous function f : [0,1]® X with f(0)=a and f(1) = b.
  • What does it mean to say X is compact?

    • X is compact if whenever {O} is a family of open subsets that cover X then there is a finite subfamily of {O} that covers X.
    Theorem (Heine- Borel): In Rn, X is compact if and only if X is closed and bounded.
  • What does it mean to say that a property of a topological space is topological?

    • If X has property P and f : X®Y is continous and onto, then Y has property P.
  • What is a topological invariant of a topological space?

    • A topological invariant is a mathematical quality (or object ) determined by a topological  spaceX so that if X and Y are homeomorphic then Y has the same mathematical quality (or object).
  • The cardinality of a topological space is a topological invariant.
  • The connectedness of a topological space is a topological invariant.
  • The compactness of a topological space is a topological invariant.
  • The Euler Characteristic of a compact surface without boundary is a topological invariant.
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    From Stanford catalog
    147. Differential Topology—Smooth manifolds, transversality, Sards’ theorem, embeddings, degree of a map, Borsuk-Ulam theorem, Hopf
    degree theorem, Jordan Curve Theorem.
    148. Algebraic Topology—Fundamental group, covering spaces, Euler characteristic, classification of surfaces, knots.