Metric and Topological Spaces

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Definition and examples of topologies

We now build on the idea of "open sets" introduced earlier.

Let X be a set. A set curlyT of subsets of X is called a topology (and the elements of curlyT are called open sets) if the following properties are satisfied.

  1. empty (the empty set), X belongs curlyT,
  2. if {Ai | i belongs I} subset curlyT then unionoveri Ai belongs curlyT,
  3. if A, B belongs curlyT then A intersect B belongs curlyT.

  1. Conditions 2. and 3. can be summarised as:
    The topology curlyT is closed under arbitrary unions and finite intersections.
  2. (X, curlyT) is called a topological space.

  1. The prototype
    Let X be any metric space and take curlyT to be the set of open sets as defined earlier. The properties verified earlier show that curlyT is a topology.

  2. Some "extremal" examples
    Take any set X and let curlyT = {empty, X}. Then curlyT is a topology called the trivial topology or indiscrete topology.
    Let X be any set and let curlyT be the set of all subsets of X. The curlyT is a topology called the discrete topology. It is the topology associated with the discrete metric.

    A topology with many open sets is called strong; one with few open sets is weak.
    The discrete topology is the strongest topology on a set, while the trivial topology is the weakest.

  3. Finite examples
    Finite sets can have many topologies on them.
    For example, Let X = {a, b} and let curlyT ={ empty, X, {a} }.
    Then curlyT is a topology called the Sierpinski topology after the Polish mathematician Waclaw Sierpinski (1882 to 1969).

    Let X = {1, 2, 3} and curlyT = {empty, {1}, {1, 2}, X}. Then curlyT is a topology.

    It is easy to check that the only metric possible on a finite set is the discrete metric. Hence these last two topologies cannot arise from a metric.

  4. The Zariski topology
    Let X be any infinite set. Define a topology on X by A belongs curlyT if X - A is finite or A = empty.
    This is called the cofinite or Zariski topology after the Belarussian mathematician Oscar Zariski (1899 to 1986)
    Examples like this are important in a subject called Algebraic Geometry.

  5. A 'different' topology on R
    Let X = R and let curlyT = {empty, R} union { (x, infinity) | x belongs R}
    Then curlyT is a topology in which, for example, the interval (0, 1) is not an open set.
    All the sets which are open in this topology are open in the usual topology. That is, this topology is weaker than the usual topology.

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JOC February 2004