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Filter (set theory)

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Fréchet filter

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).

Neighbourhood system

In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter for a point in a topological space is the collection of all neighbourhoods of Neighbourhood of a point or set An of a point (or subset) in a topological space is any open subset of that contains A is any subset that contains open neighbourhood of ; explicitly, is a neighbourhood of in if and only if there exists some open subset with . Equivalently, a neighborhood of is any set that contains in its topological interior.

Kernel (set theory)

In set theory, the kernel of a function (or equivalence kernel) may be taken to be either the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function can tell", or the corresponding partition of the domain. An unrelated notion is that of the kernel of a non-empty family of sets which by definition is the intersection of all its elements: This definition is used in the theory of filters to classify them as being free or principal.

Finite intersection property

In general topology, a branch of mathematics, a non-empty family A of subsets of a set is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of is infinite. Sets with the finite intersection property are also called centered systems and filter subbases. The finite intersection property can be used to reformulate topological compactness in terms of closed sets; this is its most prominent application.

Subbase

In topology, a subbase (or subbasis, prebase, prebasis) for a topological space with topology is a subcollection of that generates in the sense that is the smallest topology containing as open sets. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.

Pi-system

In mathematics, a pi-system (or pi-system) on a set is a collection of certain subsets of such that is non-empty. If then That is, is a non-empty family of subsets of that is closed under non-empty finite intersections. The importance of pi-systems arises from the fact that if two probability measures agree on a pi-system, then they agree on the sigma-algebra generated by that pi-system. Moreover, if other properties, such as equality of integrals, hold for the pi-system, then they hold for the generated sigma-algebra as well.

Ultrafilter on a set

In the mathematical field of set theory, an ultrafilter on a set is a maximal filter on the set In other words, it is a collection of subsets of that satisfies the definition of a filter on and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of that is also a filter. (In the above, by definition a filter on a set does not contain the empty set.) Equivalently, an ultrafilter on the set can also be characterized as a filter on with the property that for every subset of either or its complement belongs to the ultrafilter.

Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codomain of this function is usually some topological space. The motivation for generalizing the notion of a sequence is that, in the context of topology, sequences do not fully encode all information about functions between topological spaces.

Filter (mathematics)

In mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal. Special cases of filters include ultrafilters, which are filters that cannot be enlarged, and describe nonconstructive techniques in mathematical logic. Filters on sets were introduced by Henri Cartan in 1937.

Tychonoff's theorem

In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed Tychonoff), who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1935 article of Tychonoff, A.