Metric Topology

Overview


A metric is a function that is used to construct a topology

Metric


A metric on a set {% X %} is a function

{% d : X \times X \rightarrow [0,\infty] %}
such that

  • {% d(x,y) = 0 \iff x=y %}
  • {% d(x,y) = d(y,x) %}
  • {% d(x,z) \le d(x,y) + d(y,z) %}

Metric Space


A metric space is a set {% X %} with a metric. That is, it is a pair ({% X %}, {% d %})

Once given a metric space, one can define a ball {% B(x,\epsilon) %} to be the set of points,
{% B(x,\epsilon) = \{y|y \in X \, and \, d(x,y)<\epsilon\} %}

Metric Topology


A topology induced by a metric is a topology such that a set is deemed open iff for every point x in the set, there exists an epsilon such that the ball {% B(x,\epsilon) %} is contained within the set.

Example Metrics


  • {% L_1 %} distance
    {% d(x,y) = \sum_i |x_i-y_i| %}
  • {% L_2 %} distance (Euclidean distance)
    {% d(x,y) = \sqrt{ \sum_i (x_i-y_i)^2 } %}

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