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In mathematics, the possible topologies 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 (alt. stronger or larger) topology than τ2, or, synonymously, that τ2 is a coarser (alt. weaker or smaller) topology than τ1.
NB: Be aware that there are some authors, esp. analysts, who use the terms weak and strong with opposite meaning.
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. See topologies on the set of operators on a Hilbert space for some intricate relationships.
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