Comparison of topologies
Encyclopedia
In topology
and related areas of mathematics
comparison of topologies refers to the fact that two topological structures on a given set 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.
τ2:
.
That is, every element of τ1 is also an element of τ2. Then the topology τ1 is said to be a coarser (weaker or smaller) topology than τ2, and τ2 is said to be a finer (stronger or larger) topology than τ1.
There are some authors, especially analyst
s, who use the terms weak and strong with opposite meaning (Munkres, p. 78).
If additionally
we say τ1 is strictly coarser than τ2 and τ2 is strictly finer than τ1.
The binary relation
⊆ defines a partial ordering relation on the set of all possible topologies on X.
.
In function space
s 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.
All possible polar topologies on a dual pair
are finer than the weak topology
and coarser than the strong topology
.
Two immediate corollaries of this statement are
One can also compare topologies using neighborhood bases. Let τ1 and τ2 be two topologies on a set X and let Bi(x) be a local base for the topology τi at x ∈ X for i = 1,2. Then τ1 ⊆ τ2 if and only if for all x ∈ X, each open set U1 in B1(x) contains some open set U2 in B2(x). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.
. That is, any collection of topologies on X have a meet (or infimum
) and a join (or supremum
). The meet of a collection of topologies is the intersection
of those topologies. 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.
Every complete lattice is also a bounded lattice, which is to say that it has a greatest
and least element. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology
.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other. The set of all possible topologies on a given set forms a partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
. This order relation can be used to compare the different topologies.
Definition
Let τ1 and τ2 be two topologies on a set X such that τ1 is contained inSubset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
τ2:
.That is, every element of τ1 is also an element of τ2. Then the topology τ1 is said to be a coarser (weaker or smaller) topology than τ2, and τ2 is said to be a finer (stronger or larger) topology than τ1.
There are some authors, especially analyst
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
s, who use the terms weak and strong with opposite meaning (Munkres, p. 78).
If additionally
we say τ1 is strictly coarser than τ2 and τ2 is strictly finer than τ1.
The binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
⊆ defines a partial ordering relation on the set of all possible topologies on X.
Examples
The finest topology on X is the discrete topology. The coarsest topology on X is the trivial topologyTrivial topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such a space is sometimes called an indiscrete space, and its topology sometimes called an indiscrete topology...
.
In function space
Function space
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 many applications it is a topological space, a vector space, or both.-Examples:...
s and spaces of measures
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
there are often a number of possible topologies. See topologies on the set of operators on a Hilbert space for some intricate relationships.
All possible polar topologies on a dual pair
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....
are finer than the weak topology
Weak topology (polar topology)
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...
and coarser than the strong topology
Strong topology (polar topology)
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
Let τ1 and τ2 be two topologies on a set X. Then the following statements are equivalent:- τ1 ⊆ τ2
- the identity mapIdentity functionIn mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
idX : (X, τ2) → (X, τ1) is a continuous map. - the identity map idX : (X, τ1) → (X, τ2) is an open map (or, equivalently, a closed map)
Two immediate corollaries of this statement are
- A continuous map f : X → Y remains continuous if the topology on Y becomes coarser or the topology on X finer.
- An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser.
One can also compare topologies using neighborhood bases. Let τ1 and τ2 be two topologies on a set X and let Bi(x) be a local base for the topology τi at x ∈ X for i = 1,2. Then τ1 ⊆ τ2 if and only if for all x ∈ X, each open set U1 in B1(x) contains some open set U2 in B2(x). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.
Lattice of topologies
The set of all topologies on a set X together with the partial ordering relation ⊆ forms a complete latticeComplete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
. That is, any collection of topologies on X have a meet (or infimum
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
) and a join (or supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
). The meet of a collection of topologies is the intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of those topologies. The join, however, is not generally the union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of those topologies (the union of two topologies need not be a topology) but rather the topology generated by
Subbase
In topology, a subbase for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B...
the union.
Every complete lattice is also a bounded lattice, which is to say that it has a greatest
Greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
and least element. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology
Trivial topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such a space is sometimes called an indiscrete space, and its topology sometimes called an indiscrete topology...
.
See also
- Initial topologyInitial topologyIn general topology and related areas of mathematics, the initial topology on a set X, with respect to a family of functions on X, is the coarsest topology on X which makes those functions continuous.The subspace topology and product topology constructions are both special cases of initial...
, the coarsest topology on a set to make a family of mappings from that set continuous - Final topologyFinal topologyIn general topology and related areas of mathematics, the final topology on a set X, with respect to a family of functions into X, is the finest topology on X which makes those functions continuous.- Definition :Given a set X and a family of topological spaces Y_i with functionsf_i: Y_i \to Xthe...
, the finest topology on a set to make a family of mappings into that set continuous