Let τ1 and τ2 be two topologies on a set X such that τ1 is contained in τ2:
N.B. There are some authors, especially analysts, who use the terms weak and strong with opposite meaning.
In function spaces and spaces of measures there are often a number of possible topologies. See topologies on the set of operators on a Hilbert space for some intricate relationships.
Let τ1 and τ2 be two topologies on a set X. Then the following statements are equivalent:
Two immediate corollaries of this statement are
One can also compare topologies using neighborhood bases. Let τ1 and τ2 be two topologies on a set X and let Bi(x) be a local base for the topology τi at x ∈ X for i = 1,2. Then τ1 ⊆ τ2 if and only if for all x ∈ X, each open set U1 in B1(x) contains some open set U2 in B2(x). 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, any collection of topologies on X have a meet (or infimum) and a join (or supremum). The meet of a collection of topologies is the intersection of those topologies. The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union.
Every complete lattice is also a bounded lattice, which is to say that is has a greatest and least element. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.