Comparison of topologies

Last updated

In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.

Contents

Definition

A topology on a set may be defined as the collection of subsets which are considered to be "open". An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.

Let τ1 and τ2 be two topologies on a set X such that τ1 is contained in 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. [nb 1]

If additionally

we say τ1 is strictly coarser than τ2 and τ2 is strictly finer than τ1. [1]

The binary relation ⊆ defines a partial ordering relation on the set of all possible topologies on X.

Examples

The finest topology on X is the discrete topology; this topology makes all subsets open. The coarsest topology on X is the trivial topology; this topology only admits the empty set and the whole space as open sets.

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.

All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.

The complex vector space Cn may be equipped with either its usual (Euclidean) topology, or its Zariski topology. In the latter, a subset V of Cn is closed if and only if it consists of all solutions to some system of polynomial equations. Since any such V also is a closed set in the ordinary sense, but not vice versa, the Zariski topology is strictly weaker than the ordinary one.

Properties

Let τ1 and τ2 be two topologies on a set X. Then the following statements are equivalent:

(The identity map idX is surjective and therefore it is strongly open if and only if it is relatively open.)

Two immediate corollaries of the above equivalent statements 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 xX for i = 1,2. Then τ1τ2 if and only if for all xX, 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 lattice that is also closed under arbitrary intersections. [2] 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.

The lattice of topologies on a set is a complemented lattice; that is, given a topology on there exists a topology on such that the intersection is the trivial topology and the topology generated by the union is the discrete topology. [3] [4]

If the set has at least three elements, the lattice of topologies on is not modular, [5] and hence not distributive either.

See also

Notes

  1. There are some authors, especially analysts, who use the terms weak and strong with opposite meaning (Munkres, p. 78).

Related Research Articles

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

<span class="mw-page-title-main">Open set</span> Basic subset of a topological space

In mathematics, an open set is a generalization of an open interval in the real line.

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space. One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs.

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.

In mathematics, a base (or basis; pl.: bases) for the topology τ of a topological space (X, τ) is a family of open subsets of X such that every open set of the topology is equal to the union of some sub-family of . For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.

<span class="mw-page-title-main">General topology</span> Branch of topology

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.

In topology, a subbase for a topological space with topology is a subcollection of that generates in the sense that is the smallest topology containing as open sets. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.

In topology, an Alexandrov topology is a topology in which the intersection of every family of open sets is open. It is an axiom of topology that the intersection of every finite family of open sets is open; in Alexandrov topologies the finite restriction is dropped.

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology.

In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a vector space that is induced by the continuous dual of the vector space, by means of the bilinear form associated with the dual pair.

In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology. A topological vector space (TVS) is called a Mackey space if its topology is the same as the Mackey topology.

In general topology and related areas of mathematics, the final topology on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous.

<span class="mw-page-title-main">Locally connected space</span> Property of topological spaces

In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting of open connected sets.

In mathematics, a topological space is said to be limit point compact or weakly countably compact if every infinite subset of has a limit point in This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent.

In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic square matrix is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f : MN is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f."

In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a topological union of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion that is quite different from the notion of a weak topology generated by a set of maps.

In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements.

In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play a fundamental role in the study of distributive lattices. In particular, there is a duality between the category of Priestley spaces and the category of bounded distributive lattices.

References

  1. Munkres, James R. (2000). Topology (2nd ed.). Saddle River, NJ: Prentice Hall. pp.  77–78. ISBN   0-13-181629-2.
  2. Larson, Roland E.; Andima, Susan J. (1975). "The lattice of topologies: A survey". Rocky Mountain Journal of Mathematics. 5 (2): 177–198. doi: 10.1216/RMJ-1975-5-2-177 .
  3. Steiner, A. K. (1966). "The lattice of topologies: Structure and complementation". Transactions of the American Mathematical Society. 122 (2): 379–398. doi: 10.1090/S0002-9947-1966-0190893-2 .
  4. Van Rooij, A. C. M. (1968). "The Lattice of all Topologies is Complemented". Canadian Journal of Mathematics. 20: 805–807. doi: 10.4153/CJM-1968-079-9 .
  5. Steiner 1966, Theorem 3.1.