In
topology, a branch of
mathematics, approach spaces are a generalization of
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s, based on point-to-
set
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*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
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Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
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* Ext ...
pseudo 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
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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, given by treating ''A'' → ''A''
(0) as a
Kuratowski closure operator In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first for ...
.
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 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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''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
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Ext ...
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>.html" ;"title="'A'' �/sup>">'A'' �/sup>⊆ ''f'' 'A''sup> �/sup>.
Categorical properties
The main interest in approach spaces and their contractions is that they form a category
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with good properties, while still being quantitative like metric spaces. One can take arbitrary products, coproducts, 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 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
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.
References
*
* {{cite book , last=Lowen , first=Robert , title=Index Analysis: Approach Theory at Work , publisher=Springer , year=2015
External links
Robert Lowen
Closure operators