A Metric Space is a set with a concept of distance. That concept is embodied in a function which is called a metric. More formally, given a set, a metric on that set is a function d(x,y) that takes two elements and returns a real number. The metric has to satisfy the following conditions:

• $d(x,x)=0$
• $d(x,y)=d(y,x)$
• $d(x,y)+d(y,z){\ge}d(x,z)$
Exercise: Using the above, prove that $d(x,y){\ge}0.$

From Metric Spaces we can get the concept of Open Sets, which in turn leads to the idea of a topological space, which manages to keep the concept of closeness, without requiring the concept of distance.