View source for Lower bound of a family of topologies

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l0609301.png" /> (given on a single set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l0609302.png" />)''

The set-theoretical intersection of this family, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l0609303.png" />. It is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l0609304.png" /> and is always a topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l0609305.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l0609306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l0609307.png" /> are two topologies on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l0609308.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l0609309.png" /> is contained (as a set) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093010.png" />, then one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093011.png" />.

The topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093012.png" /> has the following property: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093013.png" /> is a topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093014.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093015.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093017.png" />. The free sum of the spaces that are obtained when all the individual topologies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093018.png" /> are put on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093019.png" /> can be mapped canonically onto the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093020.png" />. An important property of this mapping is that it is a [[Quotient mapping|quotient mapping]]. On this basis one proves general theorems on the preservation of a number of properties under the operation of intersecting topologies.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>



====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060930/l06093021.png" /> this infimum.