ULTRAFILTER Meaning and
Definition
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An ultrafilter is a concept defined within the field of mathematics, specifically in the realm of set theory and mathematical logic. It is an essential notion used to study and understand filters, which are structures that possess certain properties in the context of set theory.
In its simplest definition, an ultrafilter is a special type of filter, but with stronger properties. A filter is a collection of subsets of a given set that is closed under intersection and contains the set itself. In contrast, an ultrafilter is a filter that cannot be extended to a larger filter with the same properties.
To provide a more detailed description, an ultrafilter over a set is a collection of subsets with the following properties: (1) It is not empty and does not contain the empty set, (2) It is closed under taking supersets, (3) It is closed under finite intersections, and (4) It is maximal, meaning that it is not properly contained within any other filter satisfying the same properties.
Ultrafilters play a significant role in various branches of mathematics, including topology, logic, and algebra. They provide a way to study properties of sets and functions with respect to subsets, and they have applications in areas such as measure theory, topological dynamics, and model theory.
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A sac-like membrane of collodion, fish-bladder, or filter-paper impregnated with jels of varying condensation (in proportion to which the permeability of the filter varies); used in the ultrafiltration of colloids.
A practical medical dictionary. By Stedman, Thomas Lathrop. Published 1920.
Common Misspellings for ULTRAFILTER
- ultrafiltar
- ultrafiltgr
- ultrafilte2
- ultrafilteb
- ultrafiltez
- ultrafiltev
- ultrafiltep
- ultrafiltes
- u ltrafilter
- ul trafilter
- ult rafilter
- ultr afilter
- ultra filter
- ultraf ilter
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- ultrafilt er
- ultrafilte r
Etymology of ULTRAFILTER
The term "ultrafilter" is derived from the Latin prefix "ultra", meaning "beyond" or "on the other side", and the word "filter", which refers to a device or mechanism that separates or removes certain elements from a mixture. In the context of mathematics, an ultrafilter is a concept that goes beyond the classical notion of a filter by obeying additional properties. The term was coined by the Hungarian mathematician Béla Kerékjártó in the 1920s and has since become widely used in mathematical logic and set theory.
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