Approach Spaces

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Approach Spaces

1. Basic concepts An approach space is a set equipped with a notion of distance between points and sets. Given a set X, a function δ : X × 2X → [0, ∞] is called a distance (on X) if it satisfies the following properties: (D1) ∀x ∈ X: δ(x, {x}) = 0, (D2) ∀x ∈ X: δ(x, ∅) = ∞, (D3) ∀x ∈ X, ∀A, B ∈ 2X : δ(x, A ∪ B) = min(δ(x, A), δ(x, B)), (D4) ∀x ∈ X, ∀A ∈ 2X , ∀ε ∈ [0, ∞]: δ(x, A)
δ(x, A(ε)) + ε, where A(ε) := {x ∈ X: δ(x, A)
ε}. The pair (X, δ) is called an approach space [7]. A closely related concept is that of a κ-metric as introduced by Šˇcepin [14, 15]. A Tychonoff space X is called a κ-metrizable space if there exists a function ρ : X × RC(X) → R, where RC(X) stands for the collection of regularly closed sets in X, fulfilling the properties: (1) ∀x ∈ X, ∀F ∈ RC(X): ρ(x, F ) = 0 if and only if x ∈ F , (2) ∀x ∈ X, ∀F, F  ∈ RC(X): F ⊂ F  ⇒ ρ(x, F  )
ρ(x, F ), (3) ∀F ∈ RC(X): ρ(·, F ) is continuous, (4) ∀(Fj )j increasing totally ordered:


  Fj ρ x, cl int = inf ρ(x, Fj ). j ∈J

j ∈J

Such a function is called a κ-metric for X. The class of κ-metrizable topological spaces contains all metrizable topological spaces. Analogous ideas also appeared in the work of Nagata [11], and Naimpally and Pareek [12] where a similar concept under the name of annihilator was used. An annihilator basically is more general than a κ-metric, only fulfills the first and third properties above, and can be defined on X × C(X) where C(X) is an arbitrary collection of closed sets. It seems to have been known to J.-I. Nagata already in 1956 that a T1 -space X is metrizable if and only if it has a continuous annihilator ρ on X × 2X such that both inf{ρ(x, F ): F ∈ F } and sup{ρ(x, F ): F ∈ F } are continuous in x for an arbitrary collection F of closed sets [11]. Naimpally and Pareek characterized (semi)stratifiable, quasi-metric, developable, first-countable and Nagata spaces via supplementary continuity conditions on annihilators [13]. Much work in the field of κ-metrics (and annihilators) is aimed at finding supplementary conditions on a κ-metric or an annihilator to insure metrizability of a given space.

Thus a typical result, obtained by T. Isiwata [3], is that a κ-metrizable space which is stratifiable is metrizable. Somewhat later J. Suzuki, K. Tamano and Y. Tanaka showed that every κ-metrizable CW-complex also is metrizable [16]. Another type of result obtained in this context concerns stability of the notion of κ-metrizability. Thus in the original papers of Šˇcepin [14, 15] it is shown that any product of κmetrizable spaces is again κ-metrizable and that the open perfect image of a κ-metrizable space is again κ-metrizable. Approach spaces form the objects of a category, the morphisms of which are defined as follows. If (X, δX ) and (Y, δY ) are approach spaces and f : X → Y is a function, then f is called a contraction if for any x ∈ X and A ⊂ X we have δY (f (x), f (A))
δX (x, A). The category of approach spaces and contractions is denoted AP. As topological spaces, approach spaces can be characterized in several different ways. The two most important other characterizations are gauges and limit operators. A gauge G is a collection of extended pseudo-quasimetrics (meaning the value ∞ is permitted), which is saturated in the sense of the following condition: (G) if d is an extended pseudo-quasi-metric such that ∀x ∈ X, ∀ε > 0 and ∀ω < ∞ ∃e ∈ G such that d(x, ·) ∧ ω
e(x, ·) + ε then d ∈ G. Given a gauge the associated distance is derived via the formula: δ(x, A) := sup inf d(x, a). d∈G a∈A

A limit operator λ is a function which to each filter F on X associates a function λF : X → [0, ∞] fulfilling the conditions: (L1) ∀x ∈ X: λ(stack x)(x) = 0,  (L2) for any family (Fj )j ∈J of filters on X: λ( j ∈J Fj ) = supj ∈J λFj , (L3) for any filter F on X and any collection of filters (S(x))x∈X on X:     λ D(S, F )
λF + sup λ S(x) (x), x∈X

where stack x stands for the filter of all sets containing x and where D(S, F ) := F ∈F x∈X S(x) is the Kowalsky diagonal filter. Given a limit operator, the associated distance is derived via the formula:  δ(x, A) := inf λF (x): F an ultrafilter, A ∈ F . Contractions too can be characterized by means of gauges and limit operators. If (X, δ) and (X , δ  ) are approach

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spaces with corresponding gauges and limit operators G, G  , λ and λ and f : X → X , then f is a contraction if and only if for any filter F on X: λ (f (F )) ◦ f
λF , if and only if for any d  ∈ G  : d  ◦ (f × f ) ∈ G. 2. Some fundamental results Approach spaces, as topological spaces, form a so-called topological category. Especially, this implies that arbitrary initial and final structures exist. Given approach spaces (determined by their gauges) (Xj , Gj )j ∈J and a source   fj : X → (Xj , Gj ) j ∈J in AP, then the initial approach structure on X is determined by the smallest gauge containing the collection

B := sup dj ◦ (fj × fj ): K ∈ 2(J ), j ∈K  ∀j ∈ K: dj ∈ Gj , where 2(J ) stands for the set of all finite subsets of J . Given a topological space (X, T ), a natural approach space is associated with it by defining the distance δT : X × 2X → [0, ∞] by 0, x ∈ clT (A), δT (x, A) := ∞, x ∈ / clT (A). A function between topological spaces then is continuous if and only if it is a contraction between the associated approach spaces. The functor TOP →

AP,

(X, T ) → (X, δT ), f

→

f,

is a full embedding of TOP into AP. TOP is actually embedded as a bireflective and bicoreflective subcategory of AP. For any space (X, δ) ∈ |AP|, its TOP-bicoreflection is given by idX : (X, δ t c ) → (X, δ), where δ t c is the distance associated with the topological closure operator given by clδ (A) := {x ∈ X | δ(x, A) = 0}. Given an extended pseudometric space (X, d), a natural approach space is associated with it via the usual distance δd : X × 2X → [0, ∞]: (x, A) → inf d(x, a). a∈A

A function between extended pseudometric spaces then is nonexpansive if and only if it is a contraction between the associated approach spaces. The functor pMET∞ → AP, (X, d) → (X, δd ), f

→

f,

Spaces with richer structures

is a full embedding of pMET∞ into AP. The category pMET∞ of extended pseudometric spaces and nonexpansive functions is embedded as a bicoreflective subcategory of AP. For any space (X, δ) ∈ AP, its pMET∞ -bicoreflection is given by idX : (X, δ mc ) → (X, δ), where δ mc is the distance determined by the extended pseudometric     dδ : X × X → [0, ∞]: (x, y) → δ x, {y} ∨ δ y, {x} . pMET∞ is not embedded bireflectively, especially it is not closed under the formation of infinite products. An infinite product of extended pseudometric spaces is a “genuine” approach space, i.e., in general neither topological nor extended pseudometric. The relationship between an extended pseudometric and the underlying topology is recaptured in AP via a canonical functor, namely the TOP-bicoreflector restricted to pMET∞ . In the case of an extended pseudometric space the TOP-bicoreflection is the underlying topological space. In the same way, given an arbitrary approach space (X, δ), (X, δ t c ) is the underlying topological approach space and the topology generating (X, δ t c ) is the topology underlying δ. The following diagram clarifies the situation, where UAP, the subcategory of so-called uniform approach spaces stands for the epireflective hull of pMET∞ in AP. The horizontal arrows are embeddings and the vertical arrows are forgetful functors. pMET∞ −→ UAP −→ AP ←− TOP ↓ ↓ ↓ pmUNIF −→ UNIF −→ qUNIF ↓ ↓ ↓ pmTOP −→ CREG −→ TOP A typical concept defined in the setting of approach spaces is the measure of compactness, which puts the so-called Hausdorff measure of noncompactness in its proper setting [6]. If (X, d) is a pseudometric space and A ⊂ X then

mH (A) := inf ε ∈ R+ : ∃x1 , . . . , xn ∈ X :A⊂

n 



B(xi , ε)

i=1

is called the Hausdorff measure of noncompactness. Given an arbitrary approach space (X, δ), the measure of compactness of X is defined as µc (X) :=

sup

inf λF (x).

F ultrafilter x∈X

The idea behind this definition is that compactness in TOP means that every ultrafilter must converge and the information given by µc is based on the verification for all ultrafilters of what their “best” convergence points are. In the case of a pseudometric approach space, µc coincides with mH . If (X, δT ) is a topological approach space, then it is compact if and only if µc (X) = 0. If (X, δd ) is a pseudometric approach

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space, then it is totally bounded if and only if µc (X) = 0 and it is bounded if and only if µc (X) < ∞. If (Xj , δj )j ∈J is a family of approach spaces, then µc


 j ∈J

 Xj

= sup µc (Xj ), j ∈J

which is essentially the Tychonoff Product Theorem. Another fundamental concept is completeness. A filter F in an approach space (X, δ) is called a Cauchy filter if infx∈X λF (x) = 0 and F is said to be a convergent filter (with limit x) if λF (x) = 0, i.e., if it converges to x in the topological bicoreflection of (X, δ). An approach space (X, δ) is said to be a complete approach space if every Cauchy filter converges. Any topological approach space is complete. If (X, δd ) is a pseudometric approach space, then it is complete if and only if (X, d) is complete. An approach space is said to have a particular topological property, e.g., Hausdorff, if its underlying topology has that property. The fact that AP contains both TOP and pMET∞ as full and isomorphism-closed subcategories has as consequence that there exist natural constructions of completion and of compactification, at least for uniform approach spaces. It is beyond the scope of this text to define these constructions in full [7]. The notion of completion in AP coincides with the usual completion in the case of pseudometric approach spaces. It is an epireflection from the subcategory of Hausdorff uniform approach spaces to the subcategory of complete Hausdorff uniform approach spaces. The compactification in AP is the counterpart of the ˇ Cech–Stone compactification in TOP. For topological spaˇ ces this compactification coincides with the Cech–Stone compactification, and in general the TOP-coreflection of this compactification is the Smirnov compactification of an associated proximity space. In the special case of a (metric) Atsuji space, the topological coreflection of the compactifiˇ cation in AP coincides with the Cech–Stone compactification of the topological coreflection. This implies that, e.g., βN can be endowed with a distance which extends the usual ˇ metric on N, and which has the Cech–Stone compactification as topological coreflection, in other words, which “disˇ tancizes” the Cech–Stone topology of βN. The compactification is an epireflection from the subcategory of Hausdorff uniform approach spaces to the subcategory of compact Hausdorff uniform approach spaces.

295 do not exist, one can remain working at the numerical level, by performing the initial construction in AP. This results in a canonical approach structure which is strongly linked to the given metrics on the original spaces and which has as underlying topology the initial topology. Examples of this situation are the following. If X is a normed space then both X and its dual X∗ can be equipped with canonical approach structures with underlying topologies the weak topology and the weak* topology. Also most other topologies considered on X or X∗ , such as, e.g., the Mackey topology or any topology describing uniform convergence on a particular class of subsets, can be derived from natural approach structures. In all cases the metric coreflection is given by the norm (or the dual norm) [10]. If X is a metric space then the hyperspace of closed sets can be endowed with various natural approach structures having as underlying topology, e.g., the Wijsman topology, the Attouch–Wets topology and the proximal topology. In all these cases the metric coreflection is given by the Hausdorff metric [9]. If X is a separable metric space then the so-called weak topology on the space of measures on X can be derived from various natural approach structures constructed with specific collections of continuous real-valued functions. The metric coreflection always is the so-called L1 -metric [7]. Probabilistic Metric Spaces have underlying approach structures in exactly the same way that ordinary metric spaces have underlying topologies. The topologies underlying these approach structures are precisely the topologies which are used in the context of probabilistic metric spaces [1]. Spaces of random variables with values in a metric space have natural approach structures with topological coreflection the topology of convergence in probability and with metric coreflection the so-called metric of equality almost everywhere [7]. Finally, another direction of research in this field is of a categorical nature. Several categorically better behaved categorical hulls, such as the Cartesian closed topological hull, the extensional topological hull and the quasitopos hull of AP have recently been described [4, 5]. Monoidal closed structures are being studied and it has recently been shown by M.M. Clementino and D. Hofmann that approach spaces appear as the lax algebras of the ultrafilter monad over numerical relations [2].

References 3. Applications in other fields of mathematics Approach spaces find applications in those fields of mathematics where initial structures of metrizable topological spaces, in particular products, occur, especially if the metrizable topological spaces are endowed with canonical metrics. In those instances instead of having to work with the underlying topological structure, since in general initial metrics

[1] P. Brock and D.C. Kent, Approach spaces, limit tower spaces and probabilistic convergence spaces, Appl. Categ. Struct. 5 (1997), 99–110. [2] M.M. Clementino and D. Hofmann, Topological features of lax algebras, Appl. Categ. Struct., to appear. [3] T. Isiwata, Metrization of additive κ-metric spaces, Proc. Amer. Math. Soc. 100 (1987), 164–168.

296 [4] E. Lowen and R. Lowen, Topological quasitopos hulls of categories containing topological and metric objects, Cahiers Topologie Géom. Différentielle Catégoriques 30 (1989), 213–228. [5] E. Lowen, R. Lowen and M. Nauwelaerts, The Cartesian closed hull of the category of approach spaces, Cahiers Topologie Géom. Différentielle Catégoriques 42 (2001), 242–260. [6] R. Lowen, Kuratowski’s measure of noncompactness revisited, Quart. J. Math. Oxford 39 (1988), 235–254. [7] R. Lowen, Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford (1997). [8] R. Lowen, A topological category suited for approximation theory? J. Approximation Theory 56 (1989), 108–117. [9] R. Lowen and M. Sioen, The Wijsman and Attouch– Wets topologies on hyperspaces revisited, Topology Appl. 70 (1996), 179–197. [10] R. Lowen and M. Sioen, Approximations in functional analysis, Resultate Math. 37 (2000), 345–372.

Section E:

Spaces with richer structures

[11] J. Nagata, A survey of metrization theory II, Questions Answers Gen. Topology 10 (1992), 15–30. [12] S.A. Naimpally and C.M. Pareek, Characterizations of metric and generalized metric spaces by real valued functions, Questions Answers Gen. Topology 8 (1990), 425–439. [13] S.A. Naimpally and C.M. Pareek, Generalized metric spaces via annihilators, Questions Answers Gen. Topology 9 (1991), 203–226. [14] E.V. Šˇcepin, On topological products, groups, and a new class of spaces more general than metric spaces, Soviet Math. Dokl. 17 (1976), 152–155. [15] E.V. Šˇcepin, On κ-metrizable spaces, Math. USSR Izv. 14 (1980), 407–440. [16] J. Suzuki, K. Tamano and Y. Tanaka, κ-metrizable spaces, startifiable spaces and metrization, Proc. Amer. Math. Soc. 105 (1989), 500–509. Robert Lowen Antwerp, Belgium