Upper bound of a family of topologies

least upper bound, on a set

The topology which is the finest of all topologies on containing all topologies of the given family (cf. Comparison of topologies). A subbase of the topology is formed by the family of all subsets of which are open in at least one topology of the family .

The family of all possible topologies on the set with the operation defined above, which consists in taking the upper bound of any subfamily, and a minimal element — the trivial topology — is a complete lattice. The upper bound of a family of topologies is also known as the inductive limit of a family of topologies.

The following interpretation of the upper bound of a family of topologies is useful. Let

be the Tikhonov product of all topological spaces which result from imparting the various topologies in the family to the set . Let be the diagonal of this product, i.e. the set of all constant mappings from into (or, which is the same thing, the set of all threads (cf. Thread) for which for all ). The set is in a natural one-to-one correspondence with the set (this can be seen by projecting the set onto any of its factors). If is equipped with the topology induced from the space , and if this topology is transferred to using the correspondence mentioned above, one obtains the upper bound of the family . This interpretation of the upper bound of a family of topologies makes it possible to understand that the upper bound of any family of Hausdorff topologies is a Hausdorff topology, and the upper bound of any family of (completely) regular topologies is a (completely) regular topology. Similar statements do not apply to families of normal and paracompact topologies. However, the upper bound of a countable family of metrizable topologies (with a countable base) is a metrizable topology (with a countable base). The diagonal is, as a rule, not closed in , and for this reason the upper bound of two compact topologies is not necessarily compact.

References

[1]	J.L. Kelley, "General topology" , Springer (1975)
[2]	N. Bourbaki, "Elements of mathematics. General topology" , Springer (1989) (Translated from French)