Overview
A metric is a function that is used to construct a topology. It takes two points as inputs, and returns a real number representing the "distance" between the inputted points.
Formal Definition
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\} %}
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.