In
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, approach spaces are a generalization of
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s, based on point-to-
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
distances, instead of point-to-point distances. They were introduced by Robert Lowen in 1989, in a series of papers on approach theory between 1988 and 1995.
Definition
Given a metric space (''X'', ''d''), or more generally, an
extended pseudo
Pseudo- (from , ) is a prefix used in a number of languages, often to mark something as a fake or insincere version.
In English, the prefix is used on both nouns and adjectives. It can be considered a privative prefix specifically denoting '' ...
quasimetric (which will be abbreviated ''∞pq-metric'' here), one can define an induced map d: ''X'' × P(''X'') →
,∞by d(''x'', ''A'') =
inf. With this example in mind, a distance on ''X'' is defined to be a map ''X'' × P(''X'') →
,∞satisfying for all ''x'' in ''X'' and ''A'', ''B'' ⊆ ''X'',
#d(''x'', ) = 0,
#d(''x'', Ø) = ∞,
#d(''x'', ''A''∪''B'') = min(d(''x'', ''A''), d(''x'', ''B'')),
#For all 0 ≤ ε ≤ ∞, d(''x'', ''A'') ≤ d(''x'', ''A''
(ε)) + ε,
where we define ''A''
(ε) = .
(The "
empty infimum is positive infinity" convention is like the
nullary intersection is everything convention.)
An approach space is defined to be a pair (''X'', d) where d is a distance function on ''X''. Every approach space has a
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, given by treating ''A'' → ''A''
(0) as a
Kuratowski closure operator
Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. He worked as a professor at the University of Warsaw and at the Math ...
.
The appropriate maps between approach spaces are the ''contractions''. A map ''f'': (''X'', d) → (''Y'', e) is a contraction if e(''f''(''x''), ''f''
'A'' ≤ d(''x'', ''A'') for all ''x'' ∈ ''X'' and ''A'' ⊆ ''X''.
Examples
Every ∞pq-metric space (''X'', ''d'') can be ''distanced'' to (''X'', d), as described at the beginning of the definition.
Given a set ''X'', the ''discrete'' distance is given by d(''x'', ''A'') = 0 if ''x'' ∈ ''A'' and d(''x'', ''A'') = ∞ if ''x'' ∉ ''A''. The
induced topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology, o ...
is the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
.
Given a set ''X'', the ''indiscrete'' distance is given by d(''x'', ''A'') = 0 if ''A'' is non-empty, and d(''x'', ''A'') = ∞ if ''A'' is empty. The induced topology is the indiscrete topology.
Given a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'', a ''topological'' distance is given by d(''x'', ''A'') = 0 if ''x'' ∈
''A'', and d(''x'', ''A'') = ∞ otherwise. The induced topology is the original topology. In fact, the only two-valued distances are the topological distances.
Let P =
, ∞be the
extended non-negative
reals. Let d
+(''x'', ''A'') = max(''x'' −
sup ''A'', 0) for ''x'' ∈ P and ''A'' ⊆ P. Given any approach space (''X'', d), the maps (for each ''A'' ⊆ ''X'') d(., ''A'') : (''X'', d) → (P, d
+) are contractions.
On P, let e(''x'', ''A'') = inf for ''x'' < ∞, let e(∞, ''A'') = 0 if ''A'' is unbounded, and let e(∞, ''A'') = ∞ if ''A'' is bounded. Then (P, e) is an approach space. Topologically, P is the one-point compactification of
[0, ∞). Note that e extends the ordinary Euclidean distance. This cannot be done with the ordinary Euclidean metric.
Let βN be the Stone–Čech compactification of the integers. A point ''U'' ∈ βN is an ultrafilter on N. A subset ''A'' ⊆ βN induces a filter ''F''(''A'') = ∩ . Let b(''U'', ''A'') = sup. Then (βN, b) is an approach space that extends the ordinary Euclidean distance on N. In contrast, βN is not metrizable.
Equivalent definitions
Lowen has offered at least seven equivalent formulations. Two of them are below.
Let XPQ(''X'') denote the set of xpq-metrics on ''X''. A subfamily ''G'' of XPQ(''X'') is called a ''gauge'' if
#0 ∈ ''G'', where 0 is the zero metric, that is, 0(''x'', ''y'') = 0 for all ''x'', ''y'',
#''e'' ≤ ''d'' ∈ ''G'' implies ''e'' ∈ ''G'',
#''d'', ''e'' ∈ ''G'' implies max(''d'',''e'') ∈ ''G'' (the "max" here is the
pointwise maximum
In mathematics, the lower envelope or pointwise minimum of a finite set of functions is the pointwise minimum of the functions, the function whose value at every point is the minimum of the values of the functions in the given set. The concept of ...
),
#For all ''d'' ∈ XPQ(''X''), if for all ''x'' ∈ ''X'', ε > 0, ''N'' < ∞ there is ''e'' ∈ ''G'' such that min(''d''(''x'',''y''), ''N'') ≤ ''e''(''x'', ''y'') + ε for all ''y'', then ''d'' ∈ ''G''.
If ''G'' is a gauge on ''X'', then d(''x'',''A'') = sup : ''e'' ∈ ''G''} is a distance function on ''X''. Conversely, given a distance function d on ''X'', the set of ''e'' ∈ XPQ(''X'') such that e ≤ d is a gauge on ''X''. The two operations are inverse to each other.
A contraction ''f'': (''X'', d) → (''Y'', e) is, in terms of associated gauges ''G'' and ''H'' respectively, a map such that for all ''d'' ∈ ''H'', ''d''(''f''(.), ''f''(.)) ∈ ''G''.
A ''tower'' on ''X'' is a set of maps ''A'' → ''A''
�/sup> for ''A'' ⊆ ''X'', ε ≥ 0, satisfying for all ''A'', ''B'' ⊆ ''X'' and δ, ε ≥ 0
#''A'' ⊆ ''A'' �/sup>,
#Ø �/sup> = Ø,
#(''A'' ∪ ''B'') �/sup> = ''A'' �/sup> ∪ ''B'' �/sup>,
#''A'' �δ] ⊆ ''A'' �+δ/sup>,
#''A'' �/sup> = ∩δ>ε ''A'' �/sup>.
Given a distance d, the associated ''A'' → ''A''(ε) is a tower. Conversely, given a tower, the map d(''x'',''A'') = inf is a distance, and these two operations are inverses of each other.
A contraction ''f'':(''X'', d)→(''Y'', e) is, in terms of associated towers, a map such that for all ε ≥ 0, ''f'' �/sup>">'A'' �/sup>⊆ ''f'' 'A''sup> �/sup>.
Categorical properties
The main interest in approach spaces and their contractions is that they form a category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
with good properties, while still being quantitative like metric spaces. One can take arbitrary products
Product may refer to:
Business
* Product (business), an item that can be offered to a market to satisfy the desire or need of a customer.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
...
, coproducts
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprod ...
, and quotients, and the results appropriately generalize the corresponding results for topologies. One can even "distancize" such badly non-metrizable spaces like βN, the Stone–Čech compactification
In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a Universal property, universal map from a topological space ''X'' to a Compact space, compact Ha ...
of the integers.
Certain hyperspaces, measure spaces, and probabilistic metric spaces turn out to be naturally endowed with a distance. Applications have also been made to approximation theory
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...
.
References
*
*
External links
Robert Lowen
{{Metric spaces
Closure operators