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.
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 always the discrete topology, and the coarsest the trivial topology (the opposite relation to 'finer topology' being coarser topology).
In function spaces and spaces of measures there are often a number of possible topologies; the weak topology is often the coarsest choice.
