On Uniform Spaces with Invariant Nonstandard Hulls

Let X, Γ be a uniform space with its uniformity generated by a set of pseudo-metrics Γ. Let the symbol " " denote the usual infinitesimal relation on * X , and define a new infinitesimal relation " ≈ " by writing x ≈ y whenever * ρ(x, p) * ρ(y, p) for each ρ ∈ Γ and each p ∈ X. Let us call a uniform space X, Γ an S-space if the relations and ≈ coincide on fin(* X). In [14], we showed that every S-space whose uniformity is generated by a single pseudometric has invariant nonstandard hulls. Here we extend that result to all uniform spaces and use it to explore further properties of S-spaces, which can now be recognized as uniform spaces that have invariant nonstandard hulls.


Introduction
The notation is as in the abstract above.The concept of an S-space arises in connection with the question of how to construct the nonstandard hull (Luxemburg [7]) of a uniform space X, Γ within the framework of Internal Set Theory (Nelson [8], Vakil [13]).In [14] we showed how this construction can be carried out for the class of S-spaces.It was also shown in [14] that the class of S-spaces whose uniformity is generated by a single pseudo-metric includes those that have invariant nonstandard hulls (INH). 1   In Section 3 of the present paper, we extend our work in [14].We introduce the internal notion of a pseudo-precompact (PSPC) uniform space and prove that a standard uniform 2 Nader Vakil space is an S-space if and only if it is PSPC.Moreover, we show that the S-space property and the INH property are equivalent for all standard uniform spaces.Thus, in the notion of a PSPC space we provide a new internal characterization of the familiar INH property.The class of PSPC spaces extends the class of precompact uniform spaces, and Section 3 contains a discussion of two useful examples of non-precompact PSPC spaces.One example obtains by equipping a completely regular space X, τ that admits at least one unbounded f ∈ C(X) with the weakest uniformity U c with respect to which every f ∈ C(X) is uniformly continuous (Example 3.8).Another example is the uniform space X, U φ , where U φ is the finest compatible uniform structure on X, τ .This is proved in Theorem 3.11, which requires the additional condition that the space X, τ be devoid of discrete closed subspaces that have measurable cardinality.
Applications are presented in Section 4, which begins with the observation that the notion of a PSPC space provides a link between the theory of uniform spaces that have invariant nonstandard hulls and the theory of uniform spaces that have a unique structure.Through this link, we obtain two sufficient conditions for a uniform space to have invariant nonstandard hulls.Next we present a nonstandard proof of the fact that the uniform space X, U c is complete if X, τ is Lindelöf, thus providing a nonstandard proof of the well-known result that a completely regular space that is Lindelöf is realcompact.This leads to another application of the PSPC property through Theorem 4.7, which states that if a topological space X, τ is T 0 , regular and Lindelöf then a subset S of X is relatively compact if and only if f [S] is bounded for each f ∈ C(X).Our simple proof uses only the observation that if a complete uniform space X, U is PSPC then a subset S of X is relatively compact if and only if it is bounded.

Preliminaries
In this section we fix some notation and terminology concerning uniform structures and their nonstandard theory.We work with pairs X, Λ , where X is an infinite set and Λ is a set of pseudo-metrics on X .The uniformity generated by Λ will be denoted by U Λ .The pair X, U Λ as well as the pair X, Λ will be referred to as the uniform space generated by Λ.We also recall that if we begin with a uniformity U (rather than a set of pseudo-metrics Λ) on X then U may be regarded as the uniformity generated by the set Λ u of all the pseudo-metrics on X that are uniformly continuous on the space X × X equipped with the uniform structure U × U .The set Λ u is called the gauge of U and we have U = U Λu .If Λ denotes the set of all the uniformly continuous On Uniform Spaces with Invariant Nonstandard Hulls 3 pseudo-metrics on X, Λ then Λ ⊆ Λ.The set Λ is the gauge of U Λ and we have We refer to G as U Λ -bounded in case it is Λ-bounded.For the basic theory of uniform spaces, the reader may refer to Bourbaki [1], Page [10] or Willard [15].
As we mentioned earlier, the subject of this paper was inspired by the question of representation of nonstandard hulls within Internal Set Theory (IST).It is thus natural that we use IST (as outlined in Nelson [8] and expounded in Vakil [13]) as our nonstandard framework.Nevertheless, all arguments in this paper would be trivial to move over into the other foundational frameworks of NSA.
We assume that the reader is familiar with external and internal formulas (see Vakil [13, page 20]), the transfer axiom [13, page 35], the idealization axiom [13, page 71] and the standardization axiom [13, page 78].We recall that if S is a standard set and φ is a formula (external or internal) then, by the standardization axiom, S has a unique standard subset T whose standard elements satisfy φ.We shall have occasions to use this fact in the work ahead. 2 Given a class A by σ A we denote the class of all the standard members of A. Let X be a standard infinite set and let F be a standard filter on X .We recall that the monad of F is defined to be the class µ(F) = F∈ σ F F .Moreover, by the idealization axiom there always exists an element F ∈ F such that F ⊆ µ(F).The basic concepts and results concerning the nonstandard theory of uniform spaces are reviewed next.

Definition
Let Λ be a standard family of pseudo-metrics on an infinite standard set X .Then we write: The members of the classes (a)-(c) are called, respectively, finite, pre-nearstandard, and nearstandard.It is evident that Recall that the space X, Λ is compact if and only if X = ns Λ (X), and that X, Λ is complete if and only if ns Λ (X) = pns Λ (X) (see Luxemburg [7, Theorem 3.14.1,page 78]). 3The class µ Λ (x) is called the monad of x.Note that the class µ(U Λ ) is indeed the monad of the filter U Λ .We usually write x Λ y instead of x, y ∈ µ(U Λ ).We also note that x ∈ µ Λ (y) if and only if x Λ y, and (2.1.2) x Λ y if and only if ρ(x, y) 0 for all ρ ∈ σ Λ.
We will use the notations µ(U Λ ) and Λ interchangeably.For a standard topological space X, τ and a c ∈ σ X , the monad of c, denoted µ τ (c), is defined to be the class of all x ∈ X that belong to every standard G ∈ τ with c ∈ G.The class of nearstandard points of X, τ is defined to be the class Notice that ns τ (X) = ns Λ (X) in case τ is a uniformizable topology induced by a set of pseudo-metrics Λ.As to the the real line R, we use the notation for the usual infinitesimal relation in R, and we call a real number x limited if |x| < M for some standard M ∈ R. We call x unlimited and write |x| ∞ if it is not limited.
• x.This is called the standard part of x, and it is a standard real number if x is limited, and it is ±∞ if x is unlimited.The next theorem is well known, and we omit the proof.

Theorem
Let X, Λ be a standard uniform space.Then we have: is the standard subset of P(X) whose standard elements contain x then F(x) is a Λ-bounded ultrafilter on X .
(ii) A standard filter F on X is Λ-bounded if and only if µ(F) ⊆ fin Λ (X).

Pseudo-Precompact Spaces
The concept of a pseudo-precompact space introduced in this section is an internal notion equivalent (for standard sets) to the external notion of an S-space (see the abstract above).It is also equivalent (for standard sets) to the familiar notion of uniform spaces that have INH .This latter equivalence obtains through Theorem 3.7, which is the main theorem of this section.Interesting connections between the INH property and other properties of uniform spaces are revealed through Theorem 3.7.This and other applications of the notion of a pseudo-precompact space will be discussed in Section 4. In this section, we define the concept and provide some basic information about it.We begin by recalling the notion of a bounded filter in a uniform space.
3.1 Definition (Bounded Filters) Let X, U be a uniform space, and let F be a filter on X .We call F bounded (precompact) if it has an element that is U -bounded (U -precompact).
3.2 Definition (Coinciding on Filters) Let X be an infinite set, let F be a filter on X , and let U and V be two uniform structures on X .We say that U and V coincide on F if there is a G ∈ F such that 3.3 Definition (Anchored Dual of a Uniformity) Let X, Λ be a uniform space.
For each ρ ∈ Λ and p ∈ X , let ρ p (x, y) = |ρ(x, p) − ρ(y, p)| for all x, y ∈ X .If Λ = {ρ p : ρ ∈ Λ, p ∈ X} then we call the uniformity U Λ the anchored dual of the uniformity U Λ .In case we begin with a uniformity U on X , the anchored dual of U is the uniformity U that is generated by the set Λ u = {ρ p : ρ ∈ Λ u , p ∈ X}, where Λ u is the gauge of U .
Notice that when X, Λ is a standard uniform space we have x Λ y if and only if ρ(x, p) ρ(y, p) for all ρ ∈ σ Λ, p ∈ σ X.
It is not difficult to see that a uniformity U on X and its anchored dual U induce the same topology on X .Moreover, it follows easily from the triangle inequality that in general U ⊆ U .The equality holds in case X, U is precompact, but we are interested in the following weaker equality condition.

Definition (Pseudo-Precompactness)
We call a uniform space X, Λ pseudoprecompact if U Λ and its anchored dual U Λ coincide on all Λ-bounded ultrafilters on X .We shall abbreviate the term "pseudo-precompact" by PSPC.
In Theorem 3.7 below, we prove that a standard uniform space is PSPC if and only of it has INH .The following two results pave the way for that theorem.

Theorem
Let the notation be as in Definition 3.4.If X, Λ is standard then the two infinitesimal relations Λ and Λ coincide on pns Λ (X).
Proof Assume that X, Λ is a standard PSPC space.Fix x ∈ fin Λ (X), ρ ∈ Λ and ∈ R + , where ρ and are standard.Let F(x) be the unique standard subset of P(X) whose standard elements contain x.By Theorem 2.2, F(x) is a bounded ultrafilter.Hence, by hypothesis and the transfer axiom, there is a standard B ∈ F(x) on which U Λ and U Λ are identical.Therefore there exist a standard δ ∈ R + and a standard finite subset F = {p 1 , . . ., p n } of X such that the set Notice that U and V are standard, and that U[x] ⊆ V[x].Now let a i = • ρ(x, p i ), and let Clearly, A is a standard subset of X with x ∈ A. From the latter, it follows that A = ∅.Pick a standard point q ∈ A.
We are now ready for the main theorem of this paper.

Theorem
Let X, Λ be a standard uniform space.The following three conditions are equivalent.
(iii) The space X, Λ is an S-space.That is, the relations Λ and Λ coincide on fin Λ (X).
Proof Since the inclusion pns Λ (X) ⊆ fin Λ (X) holds in general, the implication [(i)→(ii)] is an immediate consequence of Lemma 3.6.The implication [(ii)→(iii)] is trivial in light of Theorem 3.5.To prove [(iii)→(i)], fix a standard Λ-bounded ultrafilter F on X and pick any standard ρ ∈ Λ.Then by Definition 3.2 and the transfer axiom there is a standard G ∈ F and a standard M ∈ R + with ρ(x, y) < M for all x, y ∈ G.This implies that G ⊆ fin Λ (X).Hence, by (iii), the relations Λ and Λ coincide on G. Since Λ is the monad of U Λ and Λ is the monad of U Λ , it follows that the uniformities U Λ and U Λ are identical on G. Apply the transfer axiom to see that each bounded ultrafilter has an element on which U Λ and U Λ are identical.The proof of the theorem is now complete.
Recall that a standard uniform space is precompact if and only if pns(X) = X .Hence, by Theorem 3.7 each precompact uniform space is PSPC.But the converse does not hold.The simplest example is the Euclidean space R n , where the uniformity is generated by its usual norm.Here is another example.

Example
Any completely regular space that admits at least one unbounded continuous function has a compatible uniformity that is PSPC without being precompact.
Proof Let X, τ be completely regular that admits at least one unbounded real-valued continuous function.Let C(X) be the set of all the real-valued continuous functions on X , and for each f ∈ C(X), let ρ f (x, y) = |f (x) − f (y).It is well known that the uniform structure on X that is generated by the set Λ c = {ρ f : f ∈ C(X)} is compatible with τ .To see that the space X, Λ c is pseudo-precompact, assume that it is standard, and verify that By Theorem 3.7, we are done once we show that fin Λc (X) ⊆ pns Λc (X).Fix x ∈ fin Λc (X), f ∈ σ C(X), and ∈ σ R + .Let a = • f (x), and let Then A is a standard set and it is not empty because it contains x.So A also contains a standard element p. Hence ρ f (x, p) = |f (x) − f (p)| < for some standard p ∈ X , which completes the proof that x ∈ pns Λc (X).We have thus shown that X, Λ c is pseudo-precompact.By transfer, there is a standard f ∈ C(X) that is not bounded.Hence there is an x ∈ X with f (x) unlimited.Hence, by 3.8.1 x / ∈ fin Λc (X).Therefore x / ∈ pns Λc (X).Since X, Λ c is precompact if and only if X = pns Λc (X), it follows that X, Λ c is not precompact.

Remark
Recall that a completely regular space X, τ is realcompact if and only if the uniform space X, Λ c is complete.Thus the preceding discussion provides a nonstandard proof for the fact that a completely regular space that is Lindelöf is realcompact.The usual proofs of this theorem use the zero-set machinery developed within the theory of rings of continuous functions (see eg Gillman and Jerison [4]).

Theorem
Let X, τ be a T 0 space that is regular and Lindelöf, and let S be a subset of X .Then S is relatively compact if and only if it is topologically bounded.
Proof The forward direction is obvious.To prove the converse, note that since S is topologically bounded, it is Λ c -bounded as a subset of the uniform space X, Λ c .Moreover, X is completely regular because it is regular and Lindelöf (see Pervin [11,Theorem 5.5.6, page 92 and Theorem 5.6.1, page 95]).In Example 3.8, we saw that the space X, Λ c is PSPC and, by Theorem 4.5, it is complete.So, by Theorem 4.4, S is relatively compact since it is a bounded subset of X, Λ c .