Editing Comparison of topologies

{{Short description|Mathematical exercise}}
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'''.

== Definition ==
A topology on a set may be defined as the collection of [[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 (set theory)|complement]] of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.) 

For definiteness the reader should think of a topology as the family of '''open sets''' of a topological space, since that is the standard meaning of the word "topology".

Let ''τ''<sub>1</sub> and ''τ''<sub>2</sub> be two topologies on a set ''X'' such that ''τ''<sub>1</sub> is contained in ''τ''<sub>2</sub>:
:<math>\tau_1 \subseteq \tau_2</math>.
That is, every element of ''τ''<sub>1</sub> is also an element of ''τ''<sub>2</sub>. Then the topology ''τ''<sub>1</sub> is said to be a '''coarser''' ('''weaker''' or '''smaller''') '''topology''' than ''τ''<sub>2</sub>, and  ''τ''<sub>2</sub> is said to be a '''finer''' ('''stronger''' or '''larger''') '''topology''' than ''τ''<sub>1</sub>.
<ref group="nb">There are some authors, especially [[mathematical analysis|analyst]]s, who use the terms ''weak'' and ''strong'' with opposite meaning. {{harv|Munkres|2000|p=78}}</ref>

If additionally 
:<math>\tau_1 \neq \tau_2</math>
we say ''τ''<sub>1</sub> is '''strictly coarser''' than  ''τ''<sub>2</sub> and ''τ''<sub>2</sub> is '''strictly finer''' than ''τ''<sub>1</sub>.{{sfn|Munkres|2000|pp=77-78}}

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 space]]s and spaces of [[Measure (mathematics)|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 topology|polar topologies]] on a [[dual pair]] are finer than the [[weak topology (polar topology)|weak topology]] and coarser than the [[strong topology (polar topology)|strong topology]].

The [[Complex coordinate space|complex vector space]] '''C'''<sup>''n''</sup> may be equipped with either its usual (Euclidean) topology, or its [[Zariski topology]]. In the latter, a subset ''V'' of '''C'''<sup>''n''</sup> 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''<sub>1</sub> and ''τ''<sub>2</sub> be two topologies on a set ''X''. Then the following statements are equivalent:
* ''τ''<sub>1</sub> ⊆ ''τ''<sub>2</sub>
* the [[identity function|identity map]] id<sub>X</sub> : (''X'', ''τ''<sub>2</sub>) → (''X'', ''τ''<sub>1</sub>) is a [[continuous map (topology)|continuous map]].
* the identity map id<sub>X</sub> : (''X'', ''τ''<sub>1</sub>) → (''X'', ''τ''<sub>2</sub>) is a [[open map|strongly/relatively open map]].

(The identity map id<sub>X</sub> is [[surjective function|surjective]] and therefore it is strongly open if and only if it is relatively open.)

Two immediate corollaries of the above equivalent statements 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]]s. Let ''τ''<sub>1</sub> and ''τ''<sub>2</sub> be two topologies on a set ''X'' and let ''B''<sub>''i''</sub>(''x'') be a local base for the topology ''τ''<sub>''i''</sub> at ''x'' ∈ ''X'' for ''i'' = 1,2. Then ''τ''<sub>1</sub> ⊆ ''τ''<sub>2</sub> if and only if for all ''x'' ∈ ''X'', each open set ''U''<sub>1</sub> in ''B''<sub>1</sub>(''x'') contains some open set ''U''<sub>2</sub> in ''B''<sub>2</sub>(''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.<ref>{{cite journal |last1=Larson |first1=Roland E. |last2=Andima |first2=Susan J. |title=The lattice of topologies: A survey |journal=Rocky Mountain Journal of Mathematics |date=1975 |volume=5 |issue=2 |pages=177–198 |doi=10.1216/RMJ-1975-5-2-177|doi-access=free }}</ref>  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 (set theory)|intersection]] of those topologies. The join, however, is not generally the [[union (set theory)|union]] of those topologies (the union of two topologies need not be a topology) but rather the topology [[subbase|generated by]] the union.

Every complete lattice is also a [[bounded lattice]], which is to say that it has a [[greatest element|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 <math>X</math> is a [[complemented lattice]]; that is, given a topology <math>\tau</math> on <math>X</math> there exists a topology <math>\tau'</math> on <math>X</math> such that the intersection <math>\tau\cap\tau'</math> is the trivial topology and the topology generated by the union <math>\tau\cup\tau'</math> is the discrete topology.<ref>{{cite journal |last1=Steiner |first1=A. K. |title=The lattice of topologies: Structure and complementation |journal=Transactions of the American Mathematical Society |date=1966 |volume=122 |issue=2 |pages=379–398 |doi=10.1090/S0002-9947-1966-0190893-2|doi-access=free }}</ref><ref>{{cite journal |last1=Van Rooij |first1=A. C. M. |title=The Lattice of all Topologies is Complemented |journal=Canadian Journal of Mathematics |date=1968 |volume=20 |pages=805–807 |doi=10.4153/CJM-1968-079-9|doi-access=free }}</ref>

If the set <math>X</math> has at least three elements, the lattice of topologies on <math>X</math> is not [[modular lattice|modular]],{{sfn|Steiner|1966|loc=Theorem 3.1}} and hence not [[distributive lattice|distributive]] either.

== See also ==

* [[Initial topology]], 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

== Notes ==
{{reflist|group=nb}}

==References==
{{Reflist}}
* {{Munkres Topology|2}}

[[Category:General topology]]
[[Category:Comparison (mathematical)|Topologies]]