Distance between points
Here is the result of submitting "Let d(x,y) be the distance..." to the
The parser had difficulty with "distance from x to a", so I rephrased
it with "between" plus the conjuction "and" instead.
The notion of distance is studied in mathematics in much the same way
as it studies anything else, that is, as something that satisfies some
set of rules or axioms and thus suffers the consequences of these rules.
The most general notion of distance belongs to the area of mathematics
called metric spaces.
Here, a set M is given with a function d of two variables, each a
member of the set M.
The rules are:
One of the consequences of being a metric space is having limits of
- symmetry, namely d(x, y) = d(y, x), for all x and y in
- identity, namely d(x, y)=0 if and only if x=y, for all x and y in
- and the triangle inequality, namely d(x, y) <= d(x, z) + d(z, y),
for all x, y, and z in M.
These functions map from one metric space to another.
Such functions have limits that are defined by the delta epsilon
definition presented on the main page of this web site.
Metric spaces are but one step in the process of abstracting the notion
More generally, the area of mathematics called point set topology
provides a setting in which functions have limits without the explicit
use of distance.
Here the notion of neighborhood and the notions of open and closed sets
prevail and the limit can be defined in terms of