Robustness, Scott Continuity, and Computability

Robustness is a property of system analyses, namely monotonic maps from the complete lattice of subsets of a (system's state) space to the two-point lattice. The definition of robustness requires the space to be a metric space. Robust analyses cannot discriminate between a subset of the metric space and its closure, therefore one can restrict to the complete lattice of closed subsets. When the metric space is compact, the complete lattice of closed subsets ordered by reverse inclusion is w-continuous and robust analyses are exactly the Scott continuous maps. Thus, one can also ask whether a robust analysis is computable (with respect to a countable base). The main result of this paper establishes a relation between robustness and Scott continuity, when the metric space is not compact. The key idea is to replace the metric space with a compact Hausdorff space, and relate robustness and Scott continuity by an adjunction between the complete lattice of closed subsets of the metric space and the w-continuous lattice of closed subsets of the compact Hausdorff space. We demonstrate the applicability of this result with several examples involving Banach spaces.


Introduction
The main contribution of this paper is relating robust analyses and Scott continuous maps (between ω-continuous lattices).This contribution is relevant to the broader endeavor of developing software tools for system analysis based on mathematical models.Typically, the behavior of a controlled system is given a priori, but for most systems the open system approach is insufficient as the correctness of the controlling system depends on properties of the environment.This requires to model also the environment.Software tools (for system analysis) manipulate formal descriptions.The key point of formal descriptions is their mathematical exactness.However, exactness should not be confused with precision.In particular, mathematical descriptions should make explicit known unknowns and the amount of imprecision.There are two unavoidable sources of imprecision: errors in measurements (on physical systems) and representations of continuous quantities in software tools.
The key feature of robust analyses is the ability to cope with small amounts of imprecision.On the other hand, analyses can be implemented in software tools only if they are computable.A definition of computability for effectively given domains has been proposed in [20,Definition 3.1], where the domains considered include those of interest for us, namely ω-continuous lattices.The key point of this, and similar proposals, is that computable maps (between effectively given domains) are necessarily Scott continuous.
For the benefit of readers, before outlining the main result of the paper, we review the context in which it is placed and recall related results, while keeping technicalities to a minimum.
Systems.First, one has to decide how to model systems.The simplest systems, i. e., discrete systems, can be modeled by a set S (of states) and a transition map t ∶ S → S describing the deterministic state change of the system in one step.The systems we consider are closed, i. e., they do not interact with the environment, or to put it differently, a model should account also for the environment.In this respect, it is important to model also known unknowns (for discrete systems, imprecision is not an issue).The simplest way to do this is by non-determinism, namely, a state in S is replaced by a set of states and the transition map is replaced by a transition relation T ⊆ S × S. In theoretical computer science, the pair (S, T ) is called a transition system.
Analyses.We move from the category of sets and relations to the category of complete lattices and monotonic maps.More precisely, we replace S with the complete lattice P(S) of subsets of S, and a relation T ⊆ S × S ′ with the monotonic map T * ∶ P(S) → P(S ′ ) such that: T * (S) △ = {s ′ ∃s ∶ S.T (s, s ′ )}.
We take as partial order ≤ on P(S) reverse inclusion, i. e., S ′ ≤ S ⇐⇒ S ′ ⊇ S. The rational for this choice is that a smaller set (of states) provides more information on the actual state of the system.The transition relations on S form a complete lattice P(S × S), where T ′ ≤ T means that T is more deterministic than T ′ .
Several analyses correspond to monotonic maps between complete lattices.For instance, reachability analysis for transition systems on S corresponds to the map R ∶ P(S×S)×P(S) → P(S) given by R(T, I) △ = T * (I), i. e., the set of states reachable in a finite number of steps from (a state in) I, while safety analysis corresponds to the map S ∶ P(S × S) × P(S) × P(S) → Σ in which Σ is the two-point lattice < ⊺ and S(T, I, E) = ⊺ △ ⇐⇒ T * (I) ∩ E = ∅, i. e., no bad state in E is reachable from I.
Approximation.The partial order on a complete lattice X allows (qualitative) comparisons, in particular we say that x ′ is an over-approximation of x when x ′ ≤ x.The category of complete lattices and monotonic maps is also the natural setting for abstract interpretation [6,7].More precisely, given an interpretation − of a (programming) language in a complete lattice X, one can choose another complete lattice X a related to X by an adjunction (also known as Galois connection) X a X, ⊺ γ α i. e., a pair of monotonic maps α (called abstraction) and γ (called concretization) such that x ′ ≤ a α(x) ⇐⇒ γ(x ′ ) ≤ x.In general, X a is simpler than X (e. g., X a can be finite) and allows interpretations − a of the language for computing over-approximations of − , i. e., γ( p a ) ≤ p for every (program) p in the language.
The adjunction between X a and X gives a systematic way of defining a − a from − .For instance, if the language is given by the BNF p ∶∶= c f (p), then an interpretation − in X is uniquely determined by c ∶ X and a monotonic map f ∶ X → X, and the abstract interpretation in X a (computing over-approximations) is determined by taking c a △ = α( c ) ∶ X a and f a △ = α ○ f ○ γ ∶ X a → X a .In static analysis, the choice of X a and − a is a matter of trade-offs between the cost of computing p a and the information provided by γ( p a ).Spaces.To model more complex systems, e. g., continuous or hybrid [13], one may have to replace sets with more complex spaces.For instance, in [16], hybrid systems are modelled by triples (S, F, G), where S is a Banach space, and F and G are binary relations on S-called flow and jump relation, respectively-which constrain how the system may evolve continuously (in time) and discontinuously (instantaneously).As in transition systems, the relations F and G allow to model known unknowns.Despite the increased complexity of models, it is still possible and useful to move to the category of complete lattices and monotonic maps and use it for analyses and abstract interpretations of these more complex systems.
Imprecision.In defining reachability for hybrid systems, we realized the need to cope with imprecision (see also [10]).Thus, in [16], we introduced safe and robust reachability analysis.In [17], we use metric spaces to formalize the notions of imprecision and robust analysis.In a metric space S, a level of imprecision δ > 0 means that one cannot distinguish two points s and s ′ when their distance d(s, s ′ ) is less than δ.If one considers subsets instead of points, and allows δ to become arbitrarily small, then one cannot distinguish two subsets that have the same closure, namely, ∀δ > 0.B(S, δ) = B(S, δ), where B(S, δ) is the open subset {s ′ ∃s ∶ S.d(s, s ′ ) < δ} and the closure S is the smallest closed subset containing S. Therefore, one can replace P(S) with the complete lattice C(S) of closed subsets, which is related to the former by the adjunction C(S) P(S), with γ an inclusion map and α a surjective map.This replacement is convenient, since the cardinality of C(S) can be smaller than that of P(S), e. g., when S is the real line R.
When A is robust, there is no loss of information in restricting to closed subsets, namely, there exists a unique monotonic map Thus, the focus of [16,17] is on analyses between complete lattices of closed subsets.[17] identifies sufficient (and almost necessary) conditions to ensure that every monotonic map A ∶ C(S) → C(S ′ ) has a best robust approximation, i. e., the biggest robust map ◻ R (A) ∶ C(S) → C(S ′ ) such that ◻ R (A) ≤ A in the lattice of monotonic maps with the point-wise order.
Scott continuity.We refer to [11] for the definitions of Scott continuous map, way-below relation ≪, and continuous lattice.Restricting to compact metric spaces is mathematically appealing, since in this case a monotonic map A ∶ C(S) → C(S ′ ) is robust exactly when it is Scott continuous, and the complete lattices C(S) and C(S ′ ) are ω-continuous [16].
A complete lattice X is ω-continuous when it has a countable base B, i. e., a countable subset of X such for every x ∶ X, the subset B x △ = {b ∶ B b ≪ x} is directed and x = sup B x .Moreover, by fixing an enumeration e of the base, one can define when an element x ∶ X is computable, namely, when the set {n e(n) ∈ B x } is a recursively enumerable subset of N. The notion of computable can be extended to Scott continuous maps between ω-continuous lattices because these maps, under the point-wise order, form an ω-continuous lattice.
Ideally, we would like to focus on computable analyses, but we settle for the broader class of Scott continuous analyses, since they are better behaved.For instance, every monotonic map A ∶ X → X ′ between complete lattices has a best Scott continuous approximation ◻ S (A) ∶ X → X ′ , while there is no best computable approximation of a monotonic map between (effectively given) ω-continuous lattices.
Related results.In [16], we define robustness as a property of analyses, i. e., monotonic maps A ∶ C(S 1 ) → C(S 2 ), where S i are metric spaces, and C(S i ) are the complete lattices of closed subsets of S i , ordered by reverse inclusion.In the same paper, it was proven that: • Robustness of A amounts to continuity with respect to suitable T 0 -topologies τ R (S i ) on the carrier sets of C(S i ), called robust topologies (see Definition 2.20).In general, these topologies depend on the metric structures of S i .
• When S i is compact, the topology τ R (S i ) coincides with the Scott topology τ S (S i ) on C(S i ).Thus, in this case, τ R (S i ) depends only on the topology induced by the metric structure on S i .
In particular, when both S 1 and S 2 are compact, robustness and Scott continuity are equivalent properties of A, and the complete lattices C(S i ) are ω-continuous.In [17], we prove that every analysis A ∶ C(S 1 ) → C(S 2 ) has a best robust approximation ◻ R (A), when S 2 is compact, with ◻ R (A)(C) given by ⋂{A(Cδ) δ > 0}, where the closed subset When S 1 is not compact, however, ◻ R (A) may fail to be Scott continuous, and C(S 1 ) may fail to be ω-continuous.

Motivating examples.
Examples of metric spaces that are not compact are Banach Spaces.In applications, one usually considers closed bounded subsets of Banach spaces.In finite-dimensional Banach spaces, all closed bounded subsets are compact, but this fails in the infinite-dimensional case.To motivate the need to go beyond compact subsets, we present some examples of closed bounded subsets of infinite-dimensional Banach spaces that are not compact: • Probability distributions for a system with a countable set of states form a closed bounded subset of ℓ 1 , i. e., the Banach space of sequences (x n n ∶ ω) in R ω such that ∑ n∶ω x n is bounded.
More generally, probability distributions on a measurable space (X, Σ) form a closed bounded subset of ca(Σ), i. e., the Banach space of countably additive bounded signed measures on Σ.This subset is not compact when the cardinality of Σ is infinite.
• Continuous maps from a compact Hausdorff space X to a compact interval [a, b] in R form a closed bounded subset of C(X), i. e., the Banach space of continuous maps from X to R. For instance, these maps could represent the height as a function of the position.
• Closed bounded subsets of feature spaces arising from kernel methods in machine learning [14].Usually feature spaces are Hilbert spaces, whose carrier sets consist of real-valued maps.All Hilbert spaces with a countable orthonormal base are isomorphic to ℓ 2 , i. e., the Hilbert space of sequences • Closed bounded subsets of Sobolev spaces W m,p (Ω), in which Ω ⊆ R n is an open set.These sets commonly appear in solution of partial differential equations [3].
Contribution.For simplicity, we consider analyses of the form A ∶ C(S) → C(1), with 1 denoting the onepoint metric space, although the results hold also when 1 is replaces by a compact metric space, e. g., a compact interval [a, b] of the real line.C( 1) is (isomorphic to) the two-point lattice Σ and the complete lattice C(S) → Σ is isomorphic to that of upward closed subsets of C(S), ordered by inclusion.This paper proposes a way to reconcile robustness and Scott continuity when S is not compact.The general idea is to construct an ω-continuous lattice D related to C(S) by an adjunction Therefore, given an analysis A ∶ C(S) → Σ, we can take the best Scott continuous approximation A ′ of A ○ ι * ∶ D → Σ-in fact, any Scott continuous approximation will do-and the composite map A ′ ○ ι * is guaranteed to be a robust approximation of A.
The ω-continuous lattice D that we construct is of the form C(S), where S is a compact Hausdorff space given by the limit of an ω op -chain of compact metric spaces related to S (see Theorem 3.9), and the adjunction ι * ⊢ ι * is determined by a continuous map ι ∶ S → S. Thus, by moving from S to S, we gain compactness by giving up the metric structure.
In general, S is not uniquely determined by S, although Theorem 3.10 provides some criteria to choose the ω op -chain of compact metric spaces which determines S.

Summary
The rest of the paper is organized as follows: • Section 2 contains the mathematical preliminaries, where we fix notation and definitions.We will also present some basic results, usually without proofs, unless the results are not available in textbooks, in which case we provide proofs or pointers to other papers which include the relevant proofs.Most definitions are standard or taken from other papers.The only exception is the category Top A of topological analyses (Definition 2.4).
• The main theoretical results are in Section 3.These include properties of idempotents and their splittings in a generic category A (e. g., Theorem 3.6) and the construction (Theorem 3.9) of a continuous map ι ∶ S → S relating a metric space S to a compact Hausdorff space S.
• In Section 4, we apply the results of Section 3 to several examples of S, namely: finite-dimensional Banach spaces ℓ m,p , infinite dimensional Banach spaces ℓ p (i. e., sequence spaces), and closed unit balls B p in sequence spaces.
• In Section 5, we investigate loss of precision when moving from the complete lattice C(S) to the ωcontinuous lattice C(S), when S is a closed unit ball B p .In Theorem 5.4 we characterize the closed subsets of ℓ p (with 1 < p < ∞) for which there is no loss of precisions as those that can be expressed as a non-empty intersections of finite unions of closed balls.
• We conclude the paper with some remarks and suggestions for future work in Section 6.The notation denotes a faithful forgetful functor, denotes the inclusion functor from a full subcategory, and ⊢ indicates the existence of a left adjoint to a functor.The left adjoints from top to bottom and left to right are: the Cauchy completion X (for X in NVS or Met), the Stone-Cech compactification βX (for X in Haus), the Hausdorff reflection HX (for X in Top 0 ), the Discrete topology DX (for X in Set).

Mathematical Preliminaries
In this section, we present the basic technical background-including the notation-that will be used throughout the paper.We assume standard terminology for topological and metric spaces.At times, we may refer to a structure by its carrier set.For instance, for a metric space (S, d), we may simply write 'the metric space S'.
We use both x ∈ X and x ∶ X to denote membership.A natural number is identified with the set of its predecessors, i. e., 0 = ∅ and n = {0, . . ., n − 1}, for any n ≥ 1.We write N or ω for the set of natural numbers.When the order matters ω denotes the set of natural numbers ordered by inclusion, while N denotes the set of natural numbers with the discrete order.We write (x n n ∶ ω) to denote a countable sequence, and when the indexing set is clear from the context, we just write (x n n).
The powerset of a set X is denoted by P(X), ⊆ denotes subset inclusion, and ⊂ denotes strict subset inclusion, i. e., A ⊂ B ⇐⇒ A ⊆ B ∧ A ≠ B. Similarly, the finite powerset (i.e., the set of finite subsets) of X is denoted by P f (X), and A ⊆ f B denotes that A is a finite subset of B.

Categories of Spaces
The spaces of interest for this paper are (extended) metric spaces1 and Hausdorff spaces.However, in examples we restrict to Banach spaces, and some constructions extend to arbitrary topological spaces.Table 1 summarizes the relations among the following categories of spaces.

Definition 2.1 (Categories of Spaces).
• Top 0 is the category of T 0 -topological spaces (X, τ ) and continuous maps.Haus and KH are the full sub-categories consisting of Hausdorff spaces (aka T 2 -spaces) and compact Hausdorff spaces, respectively.
• Met is the category of extended metric spaces (X, d), i. e., the distance d can be ∞, and short maps, There are other maps one can consider between (extended) metric spaces, in particular isometries, i. e., distance preserving maps.The forgetful functor U ∶ Met Haus maps a distance d on X to the T 2 -topology τ d on X generated by the open balls.CMS and KMS are the full sub-categories of Met consisting of Cauchy complete extended metric spaces and compact extended metric spaces, respectively.The objects in KMS are exactly the extended metric spaces whose underlying topological spaces are compact.
• NVS is the category of normed vector spaces (X, ⋅, +, − ) and short linear maps.The forgetful functor U ∶ NVS → Met maps a normed vector space to the metric space with (the same carrier and) distance d(x ′ , x) Ban is the full sub-category of NVS consisting of Banach spaces.The objects in Ban are exactly the normed vector spaces whose underlying metric spaces are complete.
The following theorems recall some properties of categories and functors in Table 1.The notation denotes a faithful forgetful functor, denotes the inclusion functor from a full sub-category, and ⊢ indicates the existence of a left adjoint to a functor.
The categories CMS, Met and Haus have also small limits and small colimits.
For the existence of sums and infinitary products it is essential to use extended metric spaces.
Theorem 2.3.The categories in the following diagram have small limits and finite sums, and the functors preserve them: KH Haus Top 0 The categories have also small colimits.

Categories of Analyses
We define an analysis as a monotonic map between complete lattices.However, we need to consider further properties of analyses, that (with the exception of computability) can be defined as continuity with respect to suitable topologies on (the carrier set of) complete lattices.For this reason, we introduce the category Top A of topological analyses, which refines the category Po A of analyses (see Table 2).

Definition 2.4 (Category of Analyses).
• Po is the category of posets and monotonic maps.
• The forgetful functor U ∶ Top 0 Po maps a T 0 -topology τ on X to the specialization order ≤ τ on • The inclusion functor A ∶ Po Top 0 maps a poset ≤ on X to the Alexandrov topology τ ≤ on X consisting of the upward closed subsets, i. e., • Po A , the category of analyses, is the full sub-category of Po consisting of complete lattices.
• Top A , the category of topological analyses, is the full sub-category of Top 0 consisting of T 0 -spaces whose specialization order is a complete lattice.
1.The categories Po and Top 0 are Po-enriched and have small limits and small colimits.
2. The functors U and A are Po-enriched, and A is left adjoint to U .
3. The functor U preserves small limits and small sums.
4. The functor A preserves finite limits and small colimits.
Proof.The Po-enrichment of Po is given by its cartesian closed structure.Since U is faithful, Thus, A is left adjoint to U also as Po-enriched functors.

The category Po
A is Po A -enriched and has small products.

The category Top
A is Po-enriched and has small products.
3. The Po-enriched functors U and A restrict to functors between Top A and Po A .
Proof.The Po A -enrichment of Po A is given by its cartesian closed structure.The other claims are easy consequences of Theorem 2.5 and the definition of Top A .
Theorem 2.7 (Topologies on a poset).Given a partial order ≤ on X, the set of T 0 -topologies on X with specialization order ≤ ordered by reverse inclusion forms a complete lattice Top(≤), where: • the least element τ is the Alexandrov topology τ ≤ .
• the top element τ ⊺ is the topology generated by the set { ↓ y y ∶ X}, where ↓ y • the nonempty sups are given by intersection.
Proof.It is easy to show that (X, τ ⊺ ) and (X, τ ) are T 0 -spaces with specialization order ≤.Given a T 0 -topology τ on X with specialization order ≤ we have τ ⊺ ⊆ τ ⊆ τ , because: Therefore, τ ⊺ and τ are respectively the top and bottom element in Top(≤).Since the topologies on a set X (ordered by reverse inclusion) form a complete lattice, with (nonempty) sups given by intersections, so do the topologies τ on X such that τ ⊺ ⊆ τ ⊆ τ .Moreover, such topologies are T 0 , because τ ⊺ is.
For every x ∶ X we have ↑ x = ⋂{ ↓ y y ∶ X ∧ x ≤ y}.When X is finite, the rhs of the equality is a finite intersection of open sets in τ ⊺ , thus ↑ x ∈ τ ⊺ , and therefore τ ⊆ τ ⊺ .
When ≤ is a complete lattice (i.e., an object in Po A ), the topologies in Top(≤) are objects in Top A .

Adjunctions and Best Approximations
A key property of categories of analyses is poset-enrichment, which provides a qualitative criterion for comparing analyses between two complete lattices, and allows to define adjunctions (aka Galois connections) between two complete lattices.

Definition 2.8 (Adjunction). An adjunction in a
The maps f and g are called left-and right adjoint, respectively.Moreover, any one of these two maps uniquely determines the other.

Remark 2.9. A characterization of adjunctions in
This characterization implies that in Po left adjoints preserve sups and (dually) right adjoints preserve infs.Given an ω op -chain A similar result holds if right adjoints are replaced by left adjoints (i.e., Po A and Top A have limits of ω op -chains of left adjoints).We recall further properties of adjunctions in Po-enriched categories (and in Po).Each of these properties has a dual, that we do not state explicitly.
In other words, if a left adjoint is mono, then it is split mono and its right adjoint is split epi.
Proof.By Remark 2.9, left adjoints in Po preserve sups, so it remains to prove the right-to-left implication.Since X is a complete lattice, g(y) The left-to-right implication follows from: The right-to-left implication follows from: Proposition 2.13.If Y is a complete lattice and X is a sub-poset of Y , then the inclusion f ∶ X → Y is a left adjoint in Po iff X is a complete lattice and sups in X are computed as in Y (i.e., f preserves sups).
Proof.The right-to-left implication follows from Proposition 2.12.For the other implication, consider the right adjoint g to f .By Remark 2.9, f preserves sups and g preserves infs.Moreover, X has all infs (i.e., is a complete lattice), because g ○ f = id X (by Proposition 2.11) and inf D = g(inf f (D)) for any subset D of X.
A subset X of a poset Y can be identified with the sub-poset of Y with carrier X and the partial order inherited from Y .Then, the inclusion f ∶ X → Y is a left adjoint in Po exactly when every y ∶ Y has a best X-approximation, and the right adjoint to f maps y to its best X-approximation.When Y is a complete lattice, Proposition 2.13 characterizes the subsets X of Y for which every y ∶ Y has a best X-approximation.
We are interested in best X-approximations of analyses in , where ≤ i is the specialization order of the topology τ i , i. e., ≤ i = U (τ i ).For existence of best X-approximations, the poset Top A (τ 1 , τ 2 ) must be a complete lattice with sups computed as in Po A (≤ 1 , ≤ 2 ).

These characterizations imply that Top
and that these subsets are closed with respect to sups computed in Po A (≤ 1 , ≤ 2 ).Therefore, every analysis f ∶ Po A (≤ 1 , ≤ 2 ) has a best Scott-continuous approximation ◻ S (f ), and a best ω-continuous approximation We introduce two sub-categories of Po A , related to the example above, which can be viewed also as full sub-categories of Top A , by mapping a complete lattice with order ≤ to the Scott topology τ S (τ ).Definition 2.16 (Category of Continuous Lattices [12]).
• CL is the category of continuous lattices, i. e., every element x in the lattice is the sup of the directed set formed by the elements way-below x, and Scott continuous maps.
• ωCL is the full sub-category of CL whose objects are ω-continuous lattices, i. e., continuous lattices with a countable subset B (called a base) such that every element x in the lattice is the sup of the directed set formed by the elements in the base way-below x.
Proposition 2.17.The Po-enriched categories in the following diagram have finite products and limits of ω op -chains of right adjoints and the functors preserve them: Moreover, the categories ωCL and CL have exponentials and the functor ωCL CL preserves them, and every ω-continuous map between ω-continuous lattices is necessarily Scott continuous.
Proof.For finite products in ωCL and CL, see [

From Spaces to Complete Lattices
Given a topological space S, the set of closed subsets of S, ordered by reverse inclusion, forms a complete lattice C(S), with sups given by intersection.We introduce several topologies on these complete lattices, but first we give the main properties of C as a functor from Haus to Po A .

Definition 2.18. The functor C ∶ Haus
Po A is defined as follows: • C(S) is the complete lattice of closed subsets of S under reverse inclusion.
• If f ∶ Haus(S, S ′ ), then the map C(f There is also a contravariant version, whose action on maps Definition 2.20 (Topologies on C(S) [16]).
3. For uniformity we write τ S (S) and τ A (S) for the Scott and Alexandrov topologies on C(S), and for conciseness we write A XY (S 1 , S 2 ) for the poset Top A (τ X (S 1 ), τ Y (S 2 )).
We call Upper topology what is better known as upper Vietoris topology.
1.If S ∶ Haus, then τ U (S) is in Top(≤), where ≤ is the partial order on C(S).
4. If S ∶ KMS is finite, then C(S) is finite and τ S (S) = τ A (S). Proof.
To prove Proof.Since the map f * is a left adjoint in Po A , it preserves all sups.Thus, it is Scott continuous.The map f * is Upper continuous, because for any C ∶ C(S) and Proof.From Theorem 2.21 we know that for any X ∶ KH, the lattice C(X) is continuous.The fact that for any X ∶ KMS, the lattice C(X) is ω-continuous is a straightforward consequence.It remains to show that, if f ∶ KH(X, Y ), then f * is Scott continuous.But this also follows from item 2 of Theorem 2.21 and Theorem 2.22.
Theorem 2.24.The functor C ∶ KH Po A preserves limits of ω op -chains.
Proof.Given an ω op -chain (p n ∶ X n+1 → X n n) in KH, its limit in KH (and Haus) is the sub-space X of ∏ n X n such that X = {x ∀n.x n = p n (x n+1 )}.By Theorem 2.10, the limit of (C(p n ) n) in Po A (and CL) But for spaces in KH compact subsets and closed subsets coincide, thus C(p n )(C) = p n (C), i. e., the image of C along p n .
By the universal property of D there exists a unique map φ ∶ C(X) → D in Po A such that: . Moreover, φ preserves infs, since the C(π n ) are right adjoints and preserve infs.Therefore, φ has a left adjoint We prove that φ ′ is the inverse of φ.Because of the adjunction φ ′ ⊣ φ we have: Given d ∶ D, we have (by the Axiom of Choice) ∀n.∀x ∶ d n .∃y , thus the first inclusion is an equality.
For the second inclusion, consider an x ∈ φ ′ (φ(C)), i. e., ∀n.x n ∈ π n (C), we prove that x ∈ C. Since C is closed and a base for the topology on X is given by the subsets

Main Results
Ideally, given a metric space S (or more generally, an extended metric space), we would like to find a compactification S of S such that the complete lattice C(S) is ω-continuous. 2We establish a weaker result, namely, given an ω-chain (g n n) of short idempotents (with certain additional properties) on a metric space S, we define a compact Hausdorff space S with a countable base and a continuous map ι ∶ S → S such that the monotonic map C(ι) ∶ C(S) → C(S) is continuous when C(S) is equipped with the Robust topology and C(S) is equipped with the Scott topology.In general, S is not a compactification of S, nor is it uniquely determined (up to isomorphism) by S, as it depends on the choice of (g n n).
The result above follows from Theorem 3.9, which is applicable under more general assumptions than having an ω-chain (g n n) of short idempotents.Another result, Theorem 3.10, gives sufficient conditions to ensure that ι ∶ S → S is both mono and epi, which is as close as we can get to have that S is a compactification of S.

Idempotents, Embeddings and Projections
In this section, we establish general properties of idempotents, split monos and split epis, that in this paper we call embeddings and projections, respectively.Definition 3.1 (idempotents & co).In a category A: 2. given two idempotents g 1 and g 2 on X we write

idempotents split (in A)
△ ⇐⇒ every idempotent (in A) has a splitting.
Given an e-p pair (e, p), we call e embedding and p projection.In general, e and p do not determine each other.A ep denotes the category whose arrows are e-p pairs and composition (e 2 , p 2 ) ○ (e 1 , p 1 ) is given by (e 2 ○ e 1 , p 1 ○ p 2 ).The forgetful functors E ∶ A ep A and P ∶ A ep A op map the e-p pair (e, p) to the embedding e and the projection p, respectively.
The notions above are absolute, i. e., they are preserved by functors, because they are defined only in terms of composition and identities.For instance, if F ∶ A A ′ is a functor and g is an idempotent on X in A, then F g is an idempotent on F X in A ′ .As is customary in Category Theory, a definition or result given for a generic category A can be recast in the dual category A op .Proposition 3.2 (Duality).

The definition of g
2. A pair (e, p) is an e-p pair from X to Y in A ⇐⇒ (p, e) is an e-p pair from X to Y in A op .In particular, the swap functor S ∶ A ep (A op ) ep , which maps (e, p) to (p, e), is an isomorphism of categories.

if (e, p
) is an e-p pair, then g = e ○ p is an idempotent.

if (g i i ∶ I
) is a family of idempotents on X which is jointly mono,i.e., (∀i.
then its sup is id X .

every arrow in
A ep is a mono.Furthermore, (e, p) is an isomorphism in A ep ⇐⇒ p is the inverse of e in A.
Proof.The proofs for Items 1 and 2 are straightforward.For the other items, we have: Consider an idempotent g of X which is an upper-bound of (g i i) (i.e., ∀i.g i ≤ g).Then, ∀i.g i ○g = g i , which implies g = id X , because (g i i) is jointly mono.
Item 4. To prove that every arrow (e, p) ∶ X Y in A ep is a mono it suffices to observe that in A every embedding e is mono (i.e., e ○ f 1 = e ○ f 2 ⇒ f 1 = f 2 ) and every projection p is epi (the dual of mono).If (e ′ , p ′ ) is the inverse of (e, p) in A ep , then e ′ is the inverse of e in A, thus p = p ○ e ○ e ′ = e ′ , because p ○ e = id X .Item 5. Uniqueness of (e, p) is immediate, because (e 2 , p 2 ) is a mono in A ep .For existence, define e = p 2 ○ e 1 and p = p 1 ○ e 2 .Then, from the assumption e 1 ○ p 1 = g 1 ≤ g 2 = e 2 ○ p 2 we derive: Moreover, if g n is the idempotent on X defined by the e-p pair (f n , q n ), then (g n n) is a jointly epi ω-chain of idempotents on X.
Proof.First, we define the family (h i,j ∶ X i → X j i, j ∶ ω) of maps in A (by induction on j − i ) Second, we prove (by induction on j − i ) the property ∀i, j.h i+1,j ○ e i = h i,j = p j ○ h i,j+1 .
• case i = j: immediate, since h i+1,i = p i and h i,i+1 = e i .
• case i < j: h i+1,j ○ e i = h i,j by definition of h i,j .For the other equality: 1. h i,j = by definition of h i,j 2. h i+1,j ○ e i = by IH on (i + 1, j) 3. p j ○ h i+1,j+1 ○ e i = by definition of h i,j+1 4. p j ○ h i,j+1 .
• case i > j: h i,j = p j ○ h i,j+1 by definition of h i,j .For the other equality: 1. h i+1,j ○ e i = by definition of h i+1,j 2. p j ○ h i+1,j+1 ○ e i = by IH on (i, j + 1) 3. p j ○ h i,j+1 = by definition of h i,j 4. h i,j .
The property implies that (h i,n i) is a cone in A from (e i i) to X n .Thus, there exists a unique q n ∶ X → X n such that ∀i.q n ○ f i = h i,n .In particular, q n ○ f n = h n,n = id Xn , i. e., (f n , q n ) ∶ X X n .The property implies also that q n = p n ○ q n+1 , because (f n n) is jointly epi.Thus, (( To prove uniqueness of (q n n), we use again that (f n n) is jointly epi and prove (by induction on j − i ) that ∀i, j.q ′ j ○ f i = h i,j when (q ′ n n) is a cone from X to (p n n) such that ∀n.q ′ n ○ f n = id Xn .
• case i = j: immediate, since q ′ i ○ f i = id Xi = h i,i by assumption on (q ′ n n) and definition of h i,i .
• case i < j: 1. q ′ j ○ f i = by definition of (f n n) 2. q ′ j ○ f i+1 ○ e i = by IH on (i + 1, j) 3. h i+1,j ○ e i = h i,j by definition of h i,j .
• case i > j: 3. p j ○ h i,j+1 = h i,j by definition of h i,j .
Consider the idempotents g n = f n ○ q n on X.From item 5 of Proposition 3.3, it follows that ∀n.g n ≤ g n+1 .The family (g n n) is jointly epi, because (f n n) is jointly epi (as colimit cones are jointly epi) and each q n is (split) epi.
The following is the dual of Proposition 3.4: Corollary 3.5.Given an ω-chain ((e n , p n ) ∶ X n X n+1 n) in A ep and a limit cone (q n ∶ X → X n n) in A from X to the ω op -chain (p n ∶ X n+1 → X n n), there exists a unique cone: in A ep from the ω-chain ((e n , p n ) n) to X.Moreover, if g n is the idempotent on X defined by the e-p pair (f n , q n ), then (g n n) is a jointly mono ω-chain of idempotents on X.
The following result implies existence and uniqueness of a map ι ∶ X → X in A such that ∀n.ι ○ f n = f n and ∀n.q n = q n ○ ι, where ((f n , q n ) ∶ X n X n) and ((f n , q n ) ∶ X n X n) are the cones in A ep given by Proposition 3.4 and Corollary 3.5.In general, there is no reason for ι to be mono, epi or iso, e. g.: in Set the map ι is always mono, but it may fail to be epi; in Set op the converse holds; in Haus the map ι is both mono and epi, but X may fail to be a sub-space of X.
Theorem 3.6.If (g n n ∶ ω) is an ω-chain of idempotents on X in A such that every g n has a splitting, say (f n , q n ) ∶ X n X, then there exists a unique ω-chain the unique map such that ∀n.q n = q n ○ ι, then ∀n.ι ○ f n = f n (see Corollary 3.5 for f n ).
Proof.By Item 5 of Proposition 3.3, we get that for each n ∶ ω there exists a unique e-p pair (e n , p n ) such that (f n+1 , q n+1 ) ○ (e n , p n ) = (f n , q n ), which implies that (( Item 1.We rely on the proof of Proposition 3.4, where q n is defined.Since (f i i) is jointly epi, ∀j.q j ○ ι = q j follows from ∀i, j.q j ○ ι ○ f i = q j ○ f i , or equivalently, from ∀i, j.q j ○ f i = h i,j , which we prove by induction on j − i : ) is a map in A ep and by definition of h i,i .
• case i < j: 3. h i+1,j ○ e i = h i,j by definition of h i,j .
• case i > j: 3. p j ○ h i,j+1 = h i,j by definition of h i,j .
Item 2. By duality, since it is the dual of Item 1.
Proposition 3.7.Given two cones The following result is the dual of Proposition 3.7.
Corollary 3.8.Given two cones (q i ∶ X → X i i ∶ I) and (q ′ i ∶ X ′ → X i i ∶ I) and a map ι ∶ X → X ′ such that ∀i.q ′ i ○ ι = q i , then (q i i) is jointly mono implies ι mono.

Extended Metric Spaces versus Compact Hausdorff Spaces
The following result requires to move between four categories (and we have added also Set) using four functors (where denotes a faithful functor and an inclusion of a full sub-category): All categories in the diagram have: • finite limits and finite sums (Theorem 2.2); • enough points, i. e., the global section functors3 Γ into Set are faithful; • splittings of idempotents; and all functors in the diagrams preserve finite limits and finite sums.Moreover, Haus has all small limits and small colimits (Theorem 2.3), where limits are computed as in Top, and limits (computed in Haus) of diagrams in KH are in KH.Also Met has all small limits and small colimits (Theorem 2.2), but the forgetful functor U from Met to Haus may not preserve all these (co)limits.
In applications, we start from a metric space S, then identify an ω-chain (g n n) of idempotents on S in Met, and by applying Theorem 3.6 in Haus we get a map ι ∶ S → S in Haus.The theorems below provide sufficient conditions to ensure that S is compact, ι is mono and epi, and, above all, that the complete lattice C(S) is ω-continuous and the monotonic map C(ι) is in A RS (S, S).These properties can be proved for maps ι ∶ S → S that are not necessarily obtained through Theorem 3.6, and the theorems below capture this greater generality.
If we start from an ω-chain (g n n) of idempotents on S in Met, then Theorem 3.6 provides candidates for (p n n) and (q n n) in the following theorem, because Met has splittings of idempotents.Theorem 3.9.If (p n ∶ S n+1 → S n n) is an ω op -chain in KMS, (q n n) is a cone from S to (p n n) in Met, (q n ∶ S → S n n) is a limit cone from S to (p n n) in Haus, and ι ∶ S → S is the unique map in Haus such that q n = q n ○ ι, then: 1. S is compact and has a countable base, thus C(S) is ω-continuous.

The monotonic map C(ι) is in A RS (S, S).
Proof.The Hausdorff space S can be identified with the set {s ∶ ∏ n S n ∀n.s n = p n (s n+1 )}, equipped with the coarsest topology O(S) making the maps q n (s) = s n continuous, i. e., the topology generated by the sub-base By Theorem 2.3, the limit S is in KH, because it is the limit (in Haus) of a diagram in KH.The topology on S n has a countable base τ b n , because S n is in KMS.Thus, the topology on S has a countable sub-base too, namely the set of open subsets of the form [B] n with B ∈ τ b n .By Theorem 2.21, the complete lattice C(X) is in CL and the topologies τ S (X) and τ U (X) coincide, when X ∶ KH, as in the case of S n and S. Therefore, τ S (X) is generated by and the way-below relation is given by , where K 1 ○ is the interior of K 1 .By Theorem 2.24, the continuous lattice C(S) is (isomorphic to) the limit of the ω op -chain of right adjoints (C(p n ) n) in CL (and in Po A ).To be more precise, the isomorphism is K ↦ (q n (K) n) and The sub-base of O(S) given above, i. e., the set of Using the above property, we have: • By property 1, this is implied by • By q n = q n ○ ι and the definition of [O] n , this is equivalent to: • In general, q ∶ Met(S, S ′ ) implies ∀δ > 0.∀C ∶ C(S).q(C δ ) ⊆ q(C) δ , and • Since q n ∶ Met(S, S n ) and S n ∶ KMS, for any C ∶ C(S) and If we start from an ω-chain (g n n) of idempotents on S in Met, rather than in Haus, then ((f n , q n ) n) and ((e n , p n ) n) in the following theorem consist of short maps.Moreover, families of maps that are jointly mono in Met are also jointly mono in Haus, because these categories have enough points.Finally, if an ω-chain (g n n) of idempotents on S (in the category A) is jointly mono (in A), then the identity on S is the sup of the ω-chain in the poset of idempotents on S. Theorem 3.10.Given an ω-chain (g n n) of idempotents on S in Haus, which is jointly mono, consider: • the limit cone (q n ∶ S → S n n) in Haus from S to the ω op -chain (p n ∶ S n+1 → S n n) (see Corollary 3.5); • the unique map ι ∶ S → S in Haus such that ∀n.q n = q n ○ ι.
Then, ι is both mono and epi in Haus.
Proof.If (g n n) is jointly mono, then also (q n n) is jointly mono.Therefore, ι is mono by Corollary 3.8.
If ((f n , q n ) ∶ S n S n) is the unique cone in Haus ep given by Corollary 3.5, then ∀n.ι ○ f n = f n , by Theorem 3.6.Therefore, to prove that ι is epi it suffices, by Proposition 3.7, to prove that (f n n) is jointly epi in Haus.This amounts to proving that the union of the images of the maps in (f n n) is dense in S. To do this we use the base of O(S) as in the proof of Theorem 3.9.For every s ∶ S and [O] n in the base (i.e.,

Examples
In this section, we consider examples of Banach spaces S, demonstrating how to apply the results of Section 3 to define a specific S in KH, and in which cases S is a compactification of S (a summary is given at the end of this section, see Table 3).In Section 5, we will study loss of precision when going from S to S. All examples considered in this section are sequence spaces.Hence, we recall some general definitions and fix notation.
Definition 4.1 (Uniform notation).We write R for the standard Banach space on the reals, and also for the underlying vector space, metric space, topological space, and set.
• Given a set I, we write R I for the product of I copies of the set R, which is also the carrier of the product of I copies of R in the categories of the vector spaces, extended metric spaces and Hausdorff spaces.
• Given a real number p in the interval [1, ∞), we write − I,p for the map from R I to [0, ∞] given by x I,p △ = (∑ i∶I x i p ) 1 p , and it is extended to p = ∞ by defining x I,∞ △ = sup i∶I x i .Since I is determined by x, we drop the subscript I and write x p .
• We write ℓ I,p for the Banach space with carrier the sub-space {x ∶ R I x p < ∞} of (the vector space) R I and norm − I,p .We write B I,p for the closed unit ball in ℓ I,p , whose elements are those x such that x p ≤ 1.The subset B I,p inherits from ℓ I,p the metric space structure.
• If I ⊆ J, then ℓ I,p is isomorphic (in the category of Banach spaces and short linear maps) to the sub-space of ℓ J,p with carrier {x ∀j ∶ J ∖ I.x j = 0}, and B I,p is a sub-space of B J,p (modulo the isomorphism).
We consider only countable I, specifically, either ω or a natural number m.We write ℓ p for ℓ ω,p and ℓ * ,p for the (normed vector) sub-space of ℓ p with carrier {x ∃n.∀i > n.x i = 0}.Usually ℓ * ,∞ is denoted c 00 .We write B p for B ω,p , and B * ,p for B p ∩ ℓ * ,p .Note that ℓ 0,p is trivial and ℓ 1,p = R for every p.
In the sequel, we use the following characterization of limits in Top and general properties of limits and colimits (valid in any category).
and by defining τ as the coarsest topology on X making the maps π i ∶ (X, τ ) → D i continuous.Proposition 4.3 (Limits commute with Limits).Given an I × J-diagram D ∶ I × J → A in a category A (with the relevant limits), if for each i ∶ I, (p i j ∶ X i → D i,j j ∶ J) is a limit cone for the J-diagram D(i, −) ∶ J → A, then the family (X i i ∶ I) extends canonically to an I-diagram X ∶ I → A, namely, for f ∶ i → i ′ in I, the map X f ∶ X i → X i ′ is the unique map in A such that for all j ∶ J, the following diagram commutes: Since one can exchange the role of I and J, there are two alternative ways of computing limits of I × J-diagrams, which necessarily produce canonically isomorphic results.

Banach space R
Consider the metric space R with distance d(x, y) = x − y and the ω-chain (r n n ∶ ω) such that (2) Each r n is idempotent and short, because: The image of r n is the compact sub-space R n △ = [−n, n], and the union S * of the sub-spaces be the splitting of r n through R n and p n = q n ○ f n+1 ∶ R n+1 → R n .Let (q n n) be the limit cone from S to the ω op -chain (p n n) in Haus.Then, by Theorem 3.9, S is compact, and by Theorem 3.10, the map ι ∶ R → S is both epi and mono in Haus.
We show that (q n n) is isomorphic to the cone (q n n) from R = [−∞, +∞] (the two-point compactification of R) to (p n n), where qn is the extension of q n to R mapping −∞ to −n and +∞ to +n.Let φ ∶ R → S be the unique map such that ∀n.q n = q n ○ φ, namely φ(x) △ = (q n (x) n).The map φ is a bijection (in Set), since the elements (s n n) in S satisfy one of the following disjoint properties: • ∀n.s n = −n, i. e., (s n n) = φ(−∞); • (s n n) is eventually constant.This happens when s m < m for some m ∶ ω.In this case (s n n) = φ(s m ); • ∀n.s n = +n, i. e., (s n n) = φ(+∞).
Therefore, (q n n) is a limit cone from R to (p n n) in Set.To prove that φ is an isomorphism in Top, it suffices to show that the topology on R is the coarsest topology making the maps qn continuous (Proposition 4.2).Since a base for the topology on R consists of the subsets of the form [−∞, x), (x, y) and (y, +∞] for x, y ∶ R, it suffices to show that every element in the base is of the form q−1 n (O) for some n ∶ ω and open subset O ∶ O(R n ).This is immediate by taking n such that x , y < n, and taking O of the form [−n, x), (x, y) and (y, +n], respectively.

Banach spaces ℓ m,∞ for 1 < m
Fix a natural number m > 1, consider the metric space S = ℓ m,∞ with distance d ∞ (x, y) = max i∶m d(x i , y i ), which coincides with the finite product R m in Met, and the ω-chain (g n n ∶ ω) defined by: where r n is as defined in (2).Since g n is defined pointwise, it is idempotent and short by inheritance, since 4.3 Banach spaces ℓ m,p for 1 < m and 1 ≤ p < ∞ This is a modification of Section 4.2, where we consider the metric space S = ℓ m,p that has the carrier of ℓ m,∞ = R m , but with distance We take the same g n used for ℓ m,∞ , as defined in (3).Clearly g n is idempotent, since this property does not depend on the distance, and is short also with respect to d p (again by inheritance), since Therefore, the metric space S n has the same carrier of R m n and distance d p .Since d p and d ∞ induce the same topology on R m n , we have that S for ℓ m,p and for ℓ m,∞ are equal and isomorphic to R m in KH.

Banach space ℓ ∞
Consider the metric space S = ℓ ∞ with distance d ∞ (x, y) △ = sup i∶ω d(x i , y i ), and the ω-chain (g n n ∶ ω) of maps on ℓ ∞ defined by ∀x ∶ ℓ ∞ .g n (x) Since g n is defined pointwise, it is idempotent and short by inheritance, e. g., 1.In the finite-dimensional cases, S is a dense sub-space of S in Haus, and S is a compactification of S. Therefore, every closed subset C of S is the intersection C ′ ⋂ S fore some closed subset C ′ of S. Since the distances d p and d ∞ induce the same topology on R m -the carrier set of both ℓ m,p and ℓ m,∞ -we have C(ℓ m,p ) = C(ℓ m,∞ ) for each p ∶ [1, ∞].Furthermore, S does not depend on p.Hence, it suffices to consider the cases S = ℓ m,∞ .The map ι ∶ ℓ m,∞ → ℓ m,∞ is a sub-space inclusion and ι * ○ ι * = id C(ℓm,∞) .
2. In the infinite-dimensional cases-as we will demonstrate-S is not a sub-space of S.More precisely, ι ∶ S → S is mono and epi, thus the image of ι is dense in S.However, ι is not a sub-space inclusion, thus there are closed subsets C of S which cannot be written as C ′ ⋂ S for some closed subset C ′ of S, and ι * ○ ι * is not the identity on C(S).
In what follows, we focus mainly on the case of the unit ball B p .As discussed in Sections 4.6 and 4.7, without loss of generality, we assume that the bijective map ι ∶ B p → B p is an identity.As a result, by going from B p to B p , the carrier set does not change, and we have to compare only the topologies, or equivalently C(B p ) ⊂ C(B p ).Since the left adjoint ι * is the inclusion map of C(B p ) into C(B p ), we have ι * (ι * (C)) = ι * (C).Thus, the loss of precision is measured by how bigger is ι * (C) in comparison to the closed subset C in C(B p ).
Remark 5.1.In applications, it is reasonable to restrict to closed bounded subsets of Banach spaces, i. e., closed subsets included in a ball of finite radius.The claims in this section are true for closed balls in ℓ p with center x and radius r > 0, but we state them for the paradigmatic case x = 0 and r = 1, to avoid extra parameters in notation.Note that, compact subsets of a Banach space are always closed and bounded, and in the finite-dimensional case the converse also holds.
To start, we present a positive result where there is no loss of precision.Proof.Write τ i for the topology on S i and ι for the mono S 1 S 2 which we can assume to be an inclusion between the carriers.Let ι * (τ 2 ) be the topology on the carrier of S 1 induced by τ 2 .Since ι is continuous, we have ι * (τ 2 ) ⊆ τ 1 .Therefore, if K is τ 1 -compact and U is an open cover of K in ι * (τ 2 ), then U is also an open cover of K in τ 1 .Hence, it has a finite sub-cover U 0 .Table 4 Some duals and double duals (up to iso).
Definition 5.7 (reflexive, weak, weak-*).For X ∶ NVS, the map η X ∶ X → X ′′ (a linear isometry) is defined as η X (x)(f ) △ = f (x) for x ∶ X and f ∶ X ′ .When the map η X is an iso, X is called reflexive.
1.The weak topology W X on X is the coarsest topology making each f ∈ X ′ continuous.
2. The weak-* topology W * X on X ′ is the coarsest topology on X ′ making η X (x) continuous for each x ∶ X.

For every
, is an isomorphism.Proof.Proofs of 1 and 2 are straightforward.To prove 3, by Proposition 5.6, and by 1 and 2, we may regard ξ as a function from ℓ p ′ to (ℓ p ) ′ , when p ∶ [1, ∞), or to (c 0 ) ′ , when p = ∞.The proof that ξ is an isomorphism may now be found in: The duals and double duals of relevance in this section are summarized in Table 4. Definition 5.9 (Topologies τ p , τ * p , τ p ).We define the following topologies on the carrier of ℓ p : 1. τ p is the original (or, norm) topology on ℓ p , i. e., the topology induced by the norm .p .
3. τ p is the topology on ℓ p as a subset of the compact Hausdorff space R ω .
We use the same notation for the topologies when restricted to a subset of ℓ p , such as B p .
According to Table 3, by using the notation of Definition 5.9, we have that S = ( B p , τp ) when S = ( B p , τ p ).Therefore, we obtain: Theorem 5.12.For each 1 ≤ p ≤ ∞, B p is (isomorphic to) the topological space ( B p , τ * p ). Proof.The topology τ p is Hausdorff.On the other hand, by the Banach-Alaoglu theorem (see, e. g., [19,Section 3.8]) the closed unit ball B p is weak-* compact, i. e., τ * p -compact.Therefore, by Lemma 5.11, it suffices to prove that τ p ⊆ τ * p .For i ∶ ω, consider the projections The set For all i ∶ ω, the sequence e i is in all the relevant pre-duals as specified in Definition 5.9 above.Furthermore Thus, each π i is weak-* continuous and Y ⊆ τ * P , which entails that τ p ⊆ τ * P .
Remark 5.13.As pointed out in Remark 5.1, although we present results for closed unit balls, they hold for arbitrary closed balls.In particular, Theorem 5.12 holds for closed balls in ℓ p , because they are all τ * p -compact.
Proof.Follows from Proposition 5.10 and Theorem 5.12.
Corollary 5.15.For each Proof.Since the space ℓ p for 1 < p < ∞ is reflexive, i. e., the weak and weak-* topologies on ℓ p coincide.The result follows from Corollary 5.14.
We have established all the preliminaries for presenting the proof of Theorem 5.4: Proof.(Theorem 5.4) (1) ⇒ (2) As C is assumed to be bounded, then it must be a subset of a closed ball B of finite radius.If ι * (ι * (C)) = C, then, by Corollary 5.15, C must be weakly closed.Thus, C is a weakly closed subset of (the weakly compact set) B. As a result, it is weakly compact.
According to [5], a subset of a separable reflexive space is weakly compact if and only if it is the non-empty intersection of finite unions of closed balls.Each space ℓ p for 1 < p < ∞ is separable and reflexive.Hence, the result follows.
(2) ⇒ (3) This is straightforward as every closed ball is a bounded-closed-convex subset.
(3) ⇒ (1) Assume that C is a non-empty intersection of finite unions of bounded-closed-convex subsets of ℓ p .To be precise, C = ⋂ i∶I ⋃ j∶ki C i,j with k ∶ ω I and C i,j bounded-closed-convex subset of ℓ p .
As a consequence of Hahn-Banach separation theorem, every closed and convex subset of a Banach space is weakly closed (see, e. g., [19,Theorem 3.12]).This entails that each C i,j is weakly closed.As the set of closed subsets (under any topology) are closed under finite unions and arbitrary intersections, then C itself is also weakly closed.Clearly, C is also bounded.The result now follows from Corollary 5.15.
Item (3) of Theorem 5.4 provides examples of practical importance where no loss of precision is incurred.
Example 5.16 (Sequence intervals).For each pair s, t ∶ ℓ p , we define the sequence interval [s, t] by: In general, sequence intervals are not norm-compact in ℓ p .They are, however, bounded, norm-closed, and convex.Hence, when 1 < p < ∞, there is no loss of precision over sequence intervals, or indeed, over any subset C of ℓ p which may be written as an intersection of finite unions of sequence intervals.
We prove that d * is a metric on the carrier of ℓ ∞ , which induces the topology on ℓ ∞ .
Proposition 5.17.For any closed ball B of finite radius in ℓ ∞ , the weak-* topology on B is induced by d * .
Proof.According to [19,Section 3.8(c), page 63], if X is a compact topological space and if some uniformly bounded sequence (f n n ∶ ω) of continuous real-valued functions separates points on X, then X is metrizable, with the metric d(x, y) = ∑ n∶ω 2 −n f n (x) − f n (y) .By the Banach-Alaoglu theorem, the ball B is weak-* compact.The countable set {π i i ∶ ω} of projections from ( 5) is separating on B, in the sense that: ∀x, y ∶ B. x ≠ y ⇒ ∃i ∶ ω.π i (x) ≠ π i (y).Proof.Let B be the closed ball in ℓ p centered at x 0 with radius r > 0. Let B ′ denote the closed ball in ℓ ∞ centered at x 0 with radius r > 0. The weak-* topology on B is the relative topology induced from the weak-* topology on B ′ .The claim now follows from Proposition 5.17.
By going from B p to B p , robustness with respect to the metric d p is replaced by robustness with respect to the metric d * .Indeed, inequality (8) shows that for any subset S of B p its δ-neighborhood B(S, δ) under d p is included its δ-neighborhood under d * . Proof.
2. From 1, we deduce that x ∈ ι * (C) if and only if: Thus, it suffices to prove that ( 9) and (10) are equivalent.

Concluding Remarks
The results in this paper are part of an overall study of robust maps.We have chosen the theory of ω-continuous lattices, within which computability can be studied using the framework of effectively given domains [20], and robustness can be analyzed using the Robust topology (Definition 2.20) over the lattice of closed subsets of the state space.In a related work, Edalat [8] has considered locally compact Hausdorff spaces instead of metric spaces, and has worked with the domain of compact subsets (ordered by reverse inclusion) instead of the complete lattice of closed subsets.Furthermore, he has investigated the relationship between Scott topology and the upper Vietoris topology, but has not studied robustness.The Robust topology lies in between Scott and the upper Vietoris topologies (Theorem 2.21).
The case of compact metric spaces has been studied in [16].This suffices to deal with the input space of typical machine learning systems, and the state space of common hybrid systems.In this paper, the focus has been non-compact metric spaces, for which, a novel approach has been presented based on approximation of the space via a (growing) sequence of compact metric sub-spaces.Non-compact spaces are relevant when dealing with perturbations of the model parameters of a system, e. g., perturbations of the activation function(s) of a neural network, or the flow F and jump G relations of a hybrid system (S, F, G).
We presented a detailed account of some examples, including (closed bounded subsets of) infinite-dimensional Banach spaces, and analyzed the important issue of precision, when it is retained, and when precision is lost.In particular, we have obtained a complete characterization of the closed subsets of reflexive spaces ℓ p (i. e., those with 1 < p < ∞), for which there is no loss of precision (Theorem 5.4).
All examples studied in this paper are sequence spaces.As such, studying other relevant spaces provides an immediate direction for future work.For instance, let Ω ⊆ R n be an open set.Lebesgue spaces L p (Ω) are examples for future work, which are relevant in the study of partial differential equations [3].

Theorem 2 . 2 .
The categories in the following diagram have finite limits and finite sums, and the functors preserve them: KMS CMS Met Haus

Theorem 2 . 10 .
The Po-enriched categories Po A and Top A have limits of ω op -chains of right adjoints.Proof.First, we prove the property for Po A .Given an ω op -chain (p n ∶ D n+1 → D n n) of right adjoints in Po A , its limit in Po is the subset of ∏ n D n given by D △ = {d ∀n.d n = p n (d n+1 )} with the point-wise order ≤ D .Since right adjoints preserve infs, infs in D exist and are computed point-wise.Thus, D ∶ Po A and the maps π n ∶ D → D n with π n (d) = d n form a limit cone (and preserve infs).

1 .
Given a Hausdorff space S, we write O(S) for the set of open subsets of S, C(S) for the set of closed subsets of S, and ↑ S for the set of C ∶ C(S) such that C ⊆ S, where S is a subset of S. The Upper topology τ U (S) is the topology on C(S) such that U ∶ τ U (S) △ ⇐⇒ ∀C ∶ U.∃O ∶ O(S).C ∶↑ O ⊆ U . 2. Given an extended metric space S, we write B(x, δ) ∶ O(S) for the open ball with center x ∶ S and radius δ > 0, and B(S, δ) ∶ O(S) for the union of open balls B(x, δ) with x ∶ S, where S is a subset of S. The Robust topology τ R (S) is the topology on then there exists x in C − C ′ .But every singleton is closed in S (because S ∶ Haus), thus the complement O of the singleton {x} is open and C ′ ∈↑ O ⊆ ↓ C. 2. Follows from [8, Proposition 3.3].3. Follows from [16, Lemma A.3]. 4. If S is finite, then C(S) is also finite.Hence, Top(≤) contains only one topology.Theorem 2.22.If f ∶ Haus(S, S ′ ), then f * ∶ A SS (S ′ , S) and f * ∶ A UU (S, S ′ ).

Theorem 2 .
23.The functor C ∶ Haus Po A restricted to KH factors through CL, and when restricted to KMS factors through ωCL.

3 .
(e, p) is an e-p pair from X to Y , notation (e, p) ∶ X Y △ ⇐⇒ X Y e p and p ○ e = id X .4. (e, p) is a splitting of g △ ⇐⇒ (e, p) is an e-p pair and g = e ○ p.
y).The image of g n is the compact sub-space S n △ = R m n and, once again, the union S * of the sub-spaces S n is S. From Section 4.1 and Proposition 4.3 we have that S is isomorphic to R m in KH.In fact, KH has all small limits.Thus, we can take I = m and J = ω op , and consider theI × J-diagram D ∶ I × J → KH such that D(i, n) = R n and D(i, n + 1 → n) is the map p n ∶ R n+1 → R n defined in Section 4.1.The limit S is obtained by first computing the limits R m n of I-diagrams D(−, n) and then the J-limit, while R m is obtained by first computing the limits R of the J-diagrams D(i, −) and then the I-limit.

Proposition 5 . 2 (
Compact sets).If S 1 S 2 in Haus, then compact subsets of S 1 are compact in S 2 .

Table 1
Categories of Spaces