lattice of topologies
, ordered by inclusion, is a complete lattice.
Let be the lattice of topologies on . Given , is called the common refinement of . By the proof above, this is the coarsest topology that is than each .
If is non-empty with more than one element, is also an atomic lattice. Each atom is a topology generated by one non-trivial subset of (non-trivial being non-empty and not ). The atom has the form , where .
Remark. In general, a lattice of topologies on a set is a sublattice of the lattice of topologies (mentioned above) on .
|Title||lattice of topologies|
|Date of creation||2013-03-22 16:54:42|
|Last modified on||2013-03-22 16:54:42|
|Last modified by||CWoo (3771)|