Finer topology
In mathematics, the possible structures of topological space on a given set X form a partially ordered set: if a collection τ1 of subsets of X contains each subset in a collection τ2, and these are both topologies on X, we say that τ1 is a finer topology than τ2.Equivalently, τ1 is a larger topology or stronger topology than τ2 or the opposite relation: τ2 is a coarser topology or smaller topology or weaker topology than τ1.
It is equivalent to say that the identity function on the set X, considered as a mapping from (X,τ1) to (X,τ2), is continuous. If τ1 is the finer of two topologies on X, we can say that it is easier for functions on X to be continuous mappings when we use τ1 since it allows us more open sets; and harder for functions to X to be continuous mappings.
The finest topology on X is the discrete topology. The coarsest topology on X is the trivial topology. Any two topologies on X have a meet and join, in the sense of lattice theory; the meet is the intersection, but the join is not in general the union.
In function spaces and spaces of measures there are often a number of possible topologies.
NB. Be aware that there are few books which use terms weak and strong with opposite meaning.