Metric Spaces are one of those mathematical topics that everyone intuitively understands. The best example of this is that of three dimensional Euclidean space . This serves as the basis for the intuitive concept of a “space”, and our ability to ascribe a distance between to points in three-dimensional space can be described by a distance function , or a metric. The underlying set together with the metric form what is called a metric space .
To be more mathematically precise, we make the following definition.
Definition. A metric space is a set together with a rule which associates with pair a real number such that
As an example, suppose that the underlying set and the metric coupled with this set is defined by . To verify that this indeed a metric space we must show that the four axioms are satisfied.
Claim. The mathematical structure in which is defined by is a metric space.
Proof. Let . Then by definition of and by definition of the absolute value function, we have that , so that axiom 1 is satisfied. Suppose now that the points in are equal, i.e. that . Then by definition of , we have that Conversely, suppose that By the triangle inequality we have that This implies that . Thus, condition (2.) is satisfied and hence the distance between two points in is zero if and only if the two points are the same. To prove condition (3.), let , so that . By virtue of the definition of the absolute value, we can say that
Thus, we see that the arguments of the proposed distance function is symmetric with respect to its arguments, namely any real numbers . To prove condition (4.), consider any three points , then the distance function between the points becomes
wherein we add zero in the form of adding and subtracting the point q (a very common trick in analysis). Then by the properties of the absolute value it follows that
where by the last equality follows from the definition of the distance function. Therefore, all four conditions have been satisfied, and hence by our definition of a metric space, it follows that whose distance function is defined by is indeed a metric space.