A topology on a set may be defined as the collection of subsets which are considered to be "open".
(An alternative definition is that it is the collection of subsets which are considered "closed".
These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa.
See topologies on the set of operators on a Hilbert space for some intricate relationships.
The complex vector space Cn may be equipped with either its usual (Euclidean) topology, or its Zariski topology.
In the latter, a subset V of Cn is closed if and only if it consists of all solutions to some system of polynomial equations.
Since any such V also is a closed set in the ordinary sense, but not vice versa, the Zariski topology is strictly weaker than the ordinary one.
Two immediate corollaries of the above equivalent statements are One can also compare topologies using neighborhood bases.
Intuitively, this makes sense: a finer topology should have smaller neighborhoods.
The set of all topologies on a set X together with the partial ordering relation ⊆ forms a complete lattice that is also closed under arbitrary intersections.
[2] That is, any collection of topologies on X have a meet (or infimum) and a join (or supremum).