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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the set of all possible topologies on a given set forms a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
. This order relation can be used for comparison of the topologies.


Definition

A topology on a set may be defined as the collection of
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s 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 A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
of an open set is closed and vice versa. In the following, it doesn't matter which definition is used. Let ''τ''1 and ''τ''2 be two topologies on a set ''X'' such that ''τ''1 is contained in ''τ''2: :\tau_1 \subseteq \tau_2. That is, every element of ''τ''1 is also an element of ''τ''2. Then the topology ''τ''1 is said to be a coarser (weaker or smaller) topology than ''τ''2, and ''τ''2 is said to be a finer (stronger or larger) topology than ''τ''1. There are some authors, especially analysts, who use the terms ''weak'' and ''strong'' with opposite meaning (Munkres, p. 78). If additionally :\tau_1 \neq \tau_2 we say ''τ''1 is strictly coarser than ''τ''2 and ''τ''2 is strictly finer than ''τ''1. The
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
⊆ defines a
partial ordering relation In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
on the set of all possible topologies on ''X''.


Examples

The finest topology on ''X'' is the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
; this topology makes all subsets open. The coarsest topology on ''X'' is the
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 spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
; this topology only admits the empty set and the whole space as open sets. In
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s and spaces of measures there are often a number of possible topologies. See
topologies on the set of operators on a Hilbert space In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space . Introduction Let (T_n)_ be a sequence of linear operators on the Banach space ...
for some intricate relationships. All possible polar topologies on a dual pair are finer than the
weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
and coarser than the strong topology. The complex vector space C''n'' may be equipped with either its usual (Euclidean) topology, or its
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
. In the latter, a subset ''V'' of C''n'' 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 ''τ''1 and ''τ''2 be two topologies on a set ''X''. Then the following statements are equivalent: * ''τ''1 ⊆ ''τ''2 * the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
idX : (''X'', ''τ''2) → (''X'', ''τ''1) is a
continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
. * the identity map idX : (''X'', ''τ''1) → (''X'', ''τ''2) is an
open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
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 ''τ''1 and ''τ''2 be two topologies on a set ''X'' and let ''B''''i''(''x'') be a local base for the topology ''τ''''i'' at ''x'' ∈ ''X'' for ''i'' = 1,2. Then ''τ''1 ⊆ ''τ''2 if and only if for all ''x'' ∈ ''X'', each open set ''U''1 in ''B''1(''x'') contains some open set ''U''2 in ''B''2(''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 In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
that is also closed under arbitrary intersections. That is, any collection of topologies on ''X'' have a ''meet'' (or
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
) and a ''join'' (or
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
). The meet of a collection of topologies is the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
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 mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
. In the case of topologies, the greatest element is the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
and the least element is the
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 spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
.


Notes


See also

*
Initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' t ...
, the coarsest topology on a set to make a family of mappings from that set continuous * Final topology, the finest topology on a set to make a family of mappings into that set continuous


References

{{Reflist, refs= {{cite book , last = Munkres , first = James R. , authorlink = James Munkres , title = Topology , url = https://archive.org/details/topology00munk , url-access = limited , edition = 2nd , publisher =
Prentice Hall Prentice Hall was an American major educational publisher owned by Savvas Learning Company. Prentice Hall publishes print and digital content for the 6–12 and higher-education market, and distributes its technical titles through the Safari ...
, location = Saddle River, NJ , year = 2000 , isbn = 0-13-181629-2 , pages
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General topology Topologies