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.
The topology T is the smallest topology on X containing B and is said to be generated by B.
Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.
The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.
The Zariski topology is a purely algebraically defined topology on the spectrum of a ring or an algebraic variety.
Many sets of operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.
A space carries the trivial topology if all points are "lumped together" in the sense that there are only two open sets, the empty set and the whole space.
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