Metric Spaces and the Topology of ℂ

Abstract

A metric space is a pair (X, d) where X is a set and d is a function from X × X into ℝ, called a distance function or metric, which satisfies the following conditions for x, j, and z in X:

$$\matrix{ {d\left( {x,y} \right) \ge 0} \cr {d\left( {x,y} \right) = 0\;{\rm{if}}\;{\rm{and}}\;{\rm{only}}\;x = y} \cr {d\left( {x,y} \right) = d\left( {y,x} \right)\left( {{\rm{sysmetry}}} \right)} \cr {d\left( {x,z} \right) \le \;d\left( {x,y} \right) + d\left( {y,z} \right)\left( {{\rm{triangle}}\;{\rm{inequality}}} \right)} \cr } $$

If x and r < 0 are fixed then define

$$\matrix{ {B\left( {x;r} \right) = \{ y \in X:\;d\left( {x,y} \right)\; < r\} } \cr {\bar B\left( {x;r} \right) = \{ y \in X:\;d\left( {x,y} \right)\; \le r\} .} \cr } $$

B(x; r) and B(x; r) are called the open and closed balls, respectively, with center x and radius r.