Lower bound of a family of topologies
(given on a single set
)
The set-theoretical intersection of this family, that is, . It is usually denoted by
and is always a topology on
. If
and
are two topologies on
and if
is contained (as a set) in
, then one writes
.
The topology has the following property: If
is a topology on
and if
for all
, then
. The free sum of the spaces that are obtained when all the individual topologies in
are put on
can be mapped canonically onto the space
. An important property of this mapping is that it is a quotient mapping. On this basis one proves general theorems on the preservation of a number of properties under the operation of intersecting topologies.
References
[1] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |
Comments
The article above actually defines the infimum of the family of topologies, which is a particular (the largest) lower bound for the family; a lower bound being any topology this infimum.
Lower bound of a family of topologies. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lower_bound_of_a_family_of_topologies&oldid=47717