In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set X may stand in relation to each other. The set of all possible topologies on a given set forms a partially ordered set. This order relation can be used to compare the different topologies. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...

Definition

Given a set X we define a relation ⊆ between two topologies τ₁ and τ₂ on X In mathematics, a relation is a generalization of arithmetic relations such as = and < which occur in statements such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation and relational algebra. ...

if τ₂ contains all the open sets of τ₁. This relation defines a partial ordering relation on the set of all possible topologies on X. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...

We say that τ₂ is finer (stronger or larger) topology than τ₁ and τ₁ is coarser (weaker or smaller) topology than τ₂.

If additionally

we say τ₁ is strictly coarser than τ₂ and τ₂ is strictly finer than τ₁.

Alternatively we say τ₁ is coarser than τ₂ if the identity mapping

is continuous. In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

Be aware that there are some authors, esp. analysts, who use the terms weak and strong with opposite meaning. Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ...

Examples

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 topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

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. In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... Measure can mean: To perform a measurement. ... In mathematics, the requirements of functional analysis mean there are several standard topologies which are given to the set of bounded linear operators on a Hilbert space. ...

All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology. In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a dual pair. ... In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form. ... In functional analysis and related areas of mathematics the weak topology is the coarsest polar topology, the topology with the fewest open sets, on a dual pair. ... In functional analysis and related areas of mathematics the strong topology is the finest polar topology, the topology with the most open sets, on a dual pair. ...

Properties

• Given a continuous functions f between two topological space X and Y then f stays continuous if the topology on Y becomes coarser or the topology on X finer.

• If τ₁ is coarser than τ₂ then every open (closed) set in τ₁ is open (closed) in τ₂

See also

• Initial topology, the coarsest topology on a set to make a familiy of mappings from that set continuous
• Final topology, the finest topology on a set to make a familiy of mappings into that set continuous