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What is Topology?

CARDAMOM PLAANTERS’ ASSOCIATION COLLEGE,

BODINAYAKANUR.

DEPARTMENT OF MATHEMATICS

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B.SUGUNA SELVARANI

ASSISTANT PROFESSOR,

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Solving Problems

  • Express difficult concepts in terms of ideas that are well understood

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  • Mathematics is mostly about determining the “sameness” of two ideas

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Sameness

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

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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.

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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.

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Sameness

  • Topology
    • Determine the sameness of two geometric objects

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Sameness

  • Topology
    • Determine the sameness of two geometric objects

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  • One can understand a difficult object if it is related to a well understood subject.

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Definitions

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

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Definitions

  • Topology
  • Open Set
  • Closed Set
  • Continuity
  • Homeomorphic

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Topology

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

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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

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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

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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.

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Example

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

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Topology

  • Question:

The following examples are not topologies. Why?

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Classifiying Sets

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

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Classifiying Sets

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

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  • A subset V of X is called Closed if the complement of V is in T.

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Open and Closed Sets

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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.

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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.

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Homeomorphism

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

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Homeomorphism

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

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  • Two topological spaces are the “same” or homeomorphic if there exists a homeomorphism from one space to the other.

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Homeomorphism

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

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  • Two topological spaces are the “same” or homeomorphic if there exists a homeomorphism from one space to the other.

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  • It is easier to tell that two spaces are NOT homeomorphic. Homeomoprhic spaces have certain characteristics.

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THANK YOU