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.

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.