Lower bound of a family of topologies

$ F $( given on a single set $ X $)

The set-theoretical intersection of this family, that is, $ \cap F $. It is usually denoted by $ \wedge F $ and is always a topology on $ X $. If $ {\mathcal T} _ {1} $ and $ {\mathcal T} _ {2} $ are two topologies on $ X $ and if $ {\mathcal T} _ {1} $ is contained (as a set) in $ {\mathcal T} _ {2} $, then one writes $ {\mathcal T} _ {1} \leq {\mathcal T} _ {2} $.

The topology $ \wedge F $ has the following property: If $ {\mathcal T} _ {1} $ is a topology on $ X $ and if $ {\mathcal T} _ {1} \leq {\mathcal T} $ for all $ {\mathcal T} \in F $, then $ {\mathcal T} _ {1} \leq \wedge F $. The free sum of the spaces that are obtained when all the individual topologies in $ F $ are put on $ X $ can be mapped canonically onto the space $ ( X , \wedge F ) $. 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 $ \leq $ this infimum.