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In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
and related areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the set of all possible topologies on a given set forms a
partially ordered set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. This
order relation Order theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematic ...
can be used for comparison of the topologies.


Definition

A topology on a set may be defined as the collection of
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
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 often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
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
contained in
τ2: :\tau_1 \subseteq \tau_2. 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 analysts, who use the terms ''weak'' and ''strong'' with opposite meaning (Munkres, p. 78). If additionally :\tau_1 \neq \tau_2 we say τ1 is strictly coarser than τ2 and τ2 is strictly finer than τ1. The
binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
⊆ defines a partial ordering relation on the set of all possible topologies on ''X''.


Examples

The finest topology on ''X'' is the
discrete topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
; this topology makes all subsets open. The coarsest topology on ''X'' is the
trivial topologyIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
; this topology only admits the empty set and the whole space as open sets. In
function space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s and spaces of measures there are often a number of possible topologies. See
topologies on the set of operators on a Hilbert spaceIn the mathematics, mathematical field of functional analysis there are several standard topology, topologies which are given to the algebra of bounded linear operators on a Banach space . Introduction Let (T_n)_ be a sequence of linear operator ...
for some intricate relationships. All possible polar topologies on a
dual pair In the field of functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction i ...
are finer than the
weak topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and coarser than the
strong topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. The complex vector space C''n'' may be equipped with either its usual (Euclidean) topology, or its
Zariski topology In algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained () ...
. 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 τ1 and τ2 be two topologies on a set ''X''. Then the following statements are equivalent: * τ1 ⊆ τ2 * the idX : (''X'', τ2) → (''X'', τ1) is a
continuous map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. * the identity map idX : (''X'', τ1) → (''X'', τ2) is an
open map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 base In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...
s. Let τ1 and τ2 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 τ1 ⊆ τ2 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 all elements of S, if such an element exists. Consequently, the term ''greatest low ...
) and a ''join'' (or
supremum In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

supremum
). The meet of a collection of topologies is the
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
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 A lattice is an abstract structure studied in the mathematics, mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least up ...
, which is to say that it has a greatest and
least element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. In the case of topologies, the greatest element is the
discrete topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
and the least element is the
trivial topologyIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
.


Notes


See also

*
Initial topology In general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used ...
, the coarsest topology on a set to make a family of mappings from that set continuous *
Final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a Set (mathematics), set X, with respect to a family of functions from Topological space, topological spaces into X, ...
, 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 is 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. Prentice Hall distributes its technical titles through th ...
, location = Saddle River, NJ , year = 2000 , isbn = 0-13-181629-2 , pages
77
78
General topology
Topologies s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are p ...