A topological space is a set X together with

(a collection of subsets of X) satisfying the following axioms:

The empty set and X are in

.

is closed under arbitrary union.

is closed under finite intersection.

The collection

is called a topology on X. The elements of X are usually called points, though they can be any mathematical objects. A topological space in which the points are functions is called a function space. The sets in

are called the open sets, and their complements in X are called closed sets.

Four examples and two non-examples of topologies on the three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing.

Source:

http://en.wikipedia.org/wiki/Topological_space