Editing Comparison of topologies (section)

==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.