Finer topology

Helping orphans the way you would do it

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.