Let's say we have a

metric space, consisting of a

set of points X, and a function

`d`(

`x`,

`y`) which gives the

distance between any two points

`x` and

`y` which are members of X.

A subset `V` of X is said to be an **open set** where for any member, `v`, of `V`, all the members of X which have a distance from `v` less than some positive nonzero constant are also members of `V`

That is:

For each `v` in `V`, there is some constant, C > 0 such that for all `x` in X, if `d`(`v`, `x`) < C, then `x` is also in `V`.

Or alternatively:

for all `v` in `V`, there exists r > 0 such that B_{X}(`v`, r) is a subset of `V`

(where B

_{X}(

`v`, r) is an

open ball in X with radius r and centered on

`v`.)

A **closed set** in X is just the complement in X of some open set `V` which is a subset of X.

By convention, the empty set is considered to be open. (We need convention because there are no members of the empty set to apply the condition to.)

X itself is trivially an open set, and, as the complement of the empty set, it is also a closed set.

If we take X as the reals, under the usual topoogy, then the set [0,1] (all the reals between and *including* 0 and 1) is not open. In a sense, this is because 0 and 1, the *greatest lower bound* (infimum) and *least upper bound* (supremum) respectively, are both members of [0,1], which contradicts our requirement for the constant C to be nonzero. In the case of the set (0,1) (just the same as [0,1], except it *doesn't* contain 0 and 1), we can use the fact that 0 and 1 are not in the set to construct a nonzero C for any point inside the set.

Any subset of X constructed by union or (finite) intersection operations on these open sets is also open, so that you can never get to a set that isn't open by doing unions or (a finite number of) intersections on these sets.

This set of open subsets of X defines the metric topology on X.

If we have other ways of specifying subsets of X which meet this requirement of closure under union and finite intersection, they will define different topologies on it. That is, a way of specifying the open sets is pretty much all you need to define a topology on your collection of points, X. The concept of a topological space is far more general than that of a metric space - it doesn't need to have any distance function that makes sense, for example. Non-metric (or even weirder) spaces may be harder to visualise, since they correspond less well to our normal physical intuitions about "space".