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 onX to be continuous mappings when we use τ1 since it allows us more open sets; and harder for functions toX to be continuous mappings.
All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.
The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union.
In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.
These topologies are all locally convex, which implies that they are defined by a family of seminorms.
The ultraweak and ultrastrong topologies are better in some ways than the weak and strong topologies, but their definitions are more complicated, so they are usually not used unless their better properties are really needed.
The weak and strong topologies are widely used as cheap approximations to the ultraweak and ultrastrong topologies, and the remaining topologies are of little practical importance.
Feeds |
Contact