Cauchy space
Encyclopedia
In general topology
and analysis
, a Cauchy space is a generalization of metric space
s and uniform space
s for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness
in topological space
s. The category of Cauchy spaces and Cauchy continuous maps is cartesian closed, and contains the category of proximity space
s.
A Cauchy space is a set X and a collection C of proper filters in the power set P(X) such that
An element of C is called a Cauchy filter, and a map f between Cauchy spaces (X,C) and (Y,D) is Cauchy continuous if f(C)⊆D; that is, each the image of each Cauchy filter in X is Cauchy in Y.
, a concept that had earlier been studied for uniform spaces.
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
and analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, a Cauchy space is a generalization of metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s and uniform space
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
s for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness
Completeness
In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.-Logical completeness:In logic, semantic completeness is the converse of soundness for formal systems...
in topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s. The category of Cauchy spaces and Cauchy continuous maps is cartesian closed, and contains the category of proximity space
Proximity space
In topology, a proximity space is an axiomatization of notions of "nearness" that hold set-to-set, as opposed to the better known point-to-set notions that characterize topological spaces....
s.
A Cauchy space is a set X and a collection C of proper filters in the power set P(X) such that
- for each x in X, the ultrafilterUltrafilterIn the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
at x, U(x), is in C. - if F is in C, and F is a subset of G, then G is in C.
- if F and G are in C and each member of F intersects each member of G, then F ∩ G is in C.
An element of C is called a Cauchy filter, and a map f between Cauchy spaces (X,C) and (Y,D) is Cauchy continuous if f(C)⊆D; that is, each the image of each Cauchy filter in X is Cauchy in Y.
Properties and definitions
Any Cauchy space is also a convergence space, where a filter F converges to x if F∩U(x) is Cauchy. In particular, a Cauchy space carries a natural topology.Examples
- Any uniform space (hence any metric spaceMetric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
, topological vector spaceTopological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
, or topological groupTopological groupIn mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
) is a Cauchy space; see Cauchy filter for definitions.
- A lattice ordered group carries a natural Cauchy structure.
- Any directed setDirected setIn mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
A may be made into a Cauchy space by declaring a filter F to be Cauchy if, given any element n of A, there isThere Is"There Is" is the second single and tenth track from Box Car Racer's eponymous album. Guitarist and vocalist Tom DeLonge still occasionally plays a solo version of this song in concert with Angels & Airwaves. The single peaked at #32 on the U.S...
an element U of F such that U is either a singleton or a subsetSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of the tail {m | m ≥ n}. Then given any other Cauchy space X, the Cauchy-continuous functionCauchy-continuous functionIn mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces...
s from A to X are the same as the Cauchy netCauchy netIn mathematics, a Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.A net is a Cauchy net if for every entourage V there exists γ such that for all α, β ≥ γ, is a member of V. More generally, in a Cauchy space, a net is Cauchy if the filter generated by the...
s in X indexed by A. If X is completeComplete spaceIn mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
, then such a function may be extended to the completion of A, which may be written A ∪ {∞}; the value of the extension at ∞ will be the limit of the net. In the case where A is the set {1, 2, 3, …} of natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s (so that a Cauchy net indexed by A is the same as a Cauchy sequenceCauchy sequenceIn mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
), then A receives the same Cauchy structure as the metric space {1, 1/2, 1/3, …}.
Category of Cauchy spaces
The natural notion of morphism between Cauchy spaces is that of a Cauchy-continuous functionCauchy-continuous function
In mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces...
, a concept that had earlier been studied for uniform spaces.