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lattice of topologies
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(Definition)
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Let  be a set. Let  be the set of all topologies on  . We may order  by inclusion. When
 , we say that
 is finer than
 , or that
 refines
 .
Proof. Clearly  is a partially ordered set when ordered by  . Furthermore, given any family of topologies
 on  , their intersection
 also defines a topology on  . Finally, let
 's be the corresponding subbases for the
 's and let
 . Then
 generated by
 is easily seen to be the supremum of the
 's. 
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 finer 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  .
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"lattice of topologies" is owned by CWoo.
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(view preamble)
See Also: coarser
| Also defines: |
common refinement |
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Cross-references: sublattice, subset, atom, atomic lattice, supremum, generated by, intersection, partially ordered set, complete lattice, inclusion, order, topologies
There are 2 references to this entry.
This is version 5 of lattice of topologies, born on 2007-04-09, modified 2007-04-10.
Object id is 9172, canonical name is LatticeOfTopologies.
Accessed 822 times total.
Classification:
| AMS MSC: | 54A10 (General topology :: Generalities :: Several topologies on one set ) |
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Pending Errata and Addenda
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