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Concept
Fréchet filter
Summary
In mathematics, the Fréchet filter, also called the cofinite filter, on a set is a certain collection of subsets of (that is, it is a particular subset of the power set of ).
A subset of belongs to the Fréchet filter if and only if the complement of in is finite.
Any such set is said to be , which is why it is alternatively called the cofinite filter on .
The Fréchet filter is of interest in topology, where filters originated, and relates to order and lattice theory because a set's power set is a partially ordered set under set inclusion (more specifically, it forms a lattice).
The Fréchet filter is named after the French mathematician Maurice Fréchet (1878-1973), who worked in topology.
A subset of a set is said to be cofinite in if its complement in (that is, the set ) is finite.
If the empty set is allowed to be in a filter, the Fréchet filter on , denoted by is the set of all cofinite subsets of .
That is:
If is a finite set, then every cofinite subset of is necessarily not empty, so that in this case, it is not necessary to make the empty set assumption made before.
This makes a on the lattice the power set of with set inclusion, given that denotes the complement of a set in the following two conditions hold:
Intersection condition If two sets are finitely complemented in then so is their intersection, since and
Upper-set condition If a set is finitely complemented in then so are its supersets in .
If the base set is finite, then since every subset of and in particular every complement, is then finite.
This case is sometimes excluded by definition or else called the improper filter on Allowing to be finite creates a single exception to the Fréchet filter's being free and non-principal since a filter on a finite set cannot be free and a non-principal filter cannot contain any singletons as members.
If is infinite, then every member of is infinite since it is simply minus finitely many of its members.
Additionally, is infinite since one of its subsets is the set of all where
The Fréchet filter is both free and non-principal, excepting the finite case mentioned above, and is included in every free filter.
Official source
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