Editing Nonstandard analysis (section)

== {{mvar|κ}}-saturation ==
It is possible to "improve" the saturation by allowing collections of higher cardinality to be intersected. A model is {{mvar|κ}}-[[Saturated model|saturated]] if whenever <math>\{A_i\}_{i \in I}</math> is a collection of internal sets with the [[finite intersection property]] and <math>|I|\leq\kappa</math>,
::<math>\bigcap_{i \in I} A_i \neq \emptyset</math>

This is useful, for instance, in a topological space {{mvar|X}}, where we may want {{math|{{!}}2<sup>''X''</sup>{{!}}}}-saturation to ensure the intersection of a standard [[neighborhood base]] is nonempty.<ref>Salbany, S.; Todorov, T. [http://www.esi.ac.at/static/esiprpr/esi666.pdf Nonstandard Analysis in Point-Set Topology] {{Webarchive|url=https://web.archive.org/web/20210122023251/https://www.esi.ac.at/static/esiprpr/esi666.pdf |date=22 January 2021 }}. Erwing Schrodinger Institute for Mathematical Physics.</ref>

For any cardinal {{mvar|κ}}, a {{mvar|κ}}-saturated extension can be constructed.<ref>Chang, C. C.; Keisler, H. J. Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. xvi+650 pp. {{isbn|0-444-88054-2}}</ref>