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''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|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|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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. General topology" , Springer  (1989)  (Translated from French)</TD></TR></table>