topology
The word “topology” is used in two related senses. As a subject, it is the study of properties that survive continuous deformation — stretching, bending, twisting, but never cutting or gluing. As a piece of data attached to a set, a topology is the precise list of which subsets are declared open, and that list is what makes the set into a topological space.
Concretely, a topology on a set X is a family T of subsets of X that contains the empty set and X, is closed under arbitrary unions, and is closed under finite intersections. These three closure properties are the whole definition; everything else in point-set topology is built on top of them.
The asymmetry between “arbitrary unions” and only “finite intersections” is deliberate and important. Allowing arbitrary intersections would collapse the theory: in the real line the intersection of all open intervals (-1/n, 1/n) for n = 1, 2, 3, ... is the single point {0}, which is not open. The finite-intersection rule is exactly what keeps single points from being forced open.
Given any collection of subsets, there is a smallest topology containing it (the topology it generates) and a largest topology contained in the intersection of all topologies — these constructions let you build topologies from a wish-list of sets you want to be open.