Definition and Examples of Metric Spaces

In ancient times, when man learnt to count his sheeps, at the same time, for the fear of losing the sheeps in the wild in dark, he, probably, have thought how far the sheeps might have gone from his hutments. Ever since the concept of distance and its measurement has puzzled the man. However, the effort put in to understand these, have paid rich dividends. The most basic definitions in mathematics were initially understood and described using the concept of a distance. For instance, the concepts such as the limit of a function, the convergence of a sequence and continuity of a function at a point used the notion of distance for their description. The definition that follows captures the most fundamental features of the concept of distance as we know it in our daily life. These are:

1. The distance between two objects is greater than or equal to zero.

2. The distance between two objects is zero if and only if the two objects are same.

3. The distance of any object x from another object y is same as the distance of y from x.

4. If there are three objects x, y and z, then the distance between x and y is less than or equal to the sum of the distance between x and z and the distance between y and z.