# 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)
How to Cite This Entry:
Upper bound of a family of topologies. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Upper_bound_of_a_family_of_topologies&oldid=49098
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article