Upper bound of a family of topologies

least upper bound, on a set $ S $

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

The family of all possible topologies on the set $ S $ 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

$$ T = \prod \{ {( S, {\mathcal T} ) } : { {\mathcal T} \in \mathfrak M } \} $$

be the Tikhonov product of all topological spaces which result from imparting the various topologies in the family $ \mathfrak M $ to the set $ S $. Let $ S ^ {*} $ be the diagonal of this product, i.e. the set of all constant mappings from $ \mathfrak M $ into $ S $( or, which is the same thing, the set of all threads (cf. Thread) $ \{ {S } : { {\mathcal T} \in \mathfrak M } \} $ for which $ S _ {\mathcal T} = S _ { {\mathcal T} ^ \prime } $ for all $ {\mathcal T} , {\mathcal T} ^ \prime \in \mathfrak M $). The set $ S ^ {*} $ is in a natural one-to-one correspondence with the set $ S $( this can be seen by projecting the set $ T $ onto any of its factors). If $ S ^ {*} $ is equipped with the topology induced from the space $ T $, and if this topology is transferred to $ S $ using the correspondence mentioned above, one obtains the upper bound of the family $ \mathfrak M $. 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 $ S ^ {*} $ is, as a rule, not closed in $ T $, 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)