Uniform Liftings of Continuous Mappings

We investigate the question of when a continuous mapping between subspaces of nonstandard hulls has a uniform lifting.


Introduction
Let H and H be (some) nonstandard hulls of * metric spaces (M, d) and (M , d ) respectively and M a subspace of H. Following Fajardo and Keisler [6, Definition 4.14], we say that a mapping f : M → H is uniformly liftable if there exists an internal mapping φ : M → M such that for every equivalence class µ ∈ M the image φ[µ] is contained in f (µ); the mapping φ in this case is said to be a uniform lifting of f .It is easily checked that every uniformly liftable mapping is continuous.The converse is not true: Example 1.1 Let d be a {0, 1}-valued * metric on * N and let H = {{x} : x ∈ * N} be the nonstandard hull of ( * N, d).(Here and in the following N denotes the set of nonnegative integers.)It is clear that the mapping f : H → H defined by f ({x}) = {χ N (x)}, where χ N : * N → {0, 1} is the characteristic function of N, is continuous but not uniformly liftable.
By [5,Proposition 4.12] and [6,Corollary 4.8,Theorem 4.18] every continuous mapping f : M → H defined on a compact set M is uniformly liftable.More generally, we have the following result: (We assume that our nonstandard universe (V(Ξ), V( * Ξ), * ) is κ-saturated, where κ is an uncountable cardinal.)Proof Assume first that f is an isometric embedding of M into H . Let D be a dense subspace of M of cardinality < κ.For every µ ∈ D choose x µ ∈ µ and y µ ∈ f (µ).For every (µ, n) ∈ D × N let Ψ µ,n be an internal set consisting of all internal mappings ψ : H → M with the following properties: By κ-saturation there exists a mapping ψ 0 : H 0 → M which belongs to the intersection of all Ψ µ,n .Since D is dense in M, by ω 1 -saturation H 0 intersects with every µ ∈ M. Since H 0 is hyperfinite, by transfer there exists an internal mapping r : M → H 0 such that d(x, r(x)) = min{d(x, y) : y ∈ H 0 } for every x ∈ M ; it follows that x ∈ µ ∈ M implies r(x) ∈ µ.It is easily checked that φ = ψ 0 • r is a uniform lifting of some mapping g : M → H . Since g is uniformly liftable, it is continuous.Mappings f and g coincide on D, and since D is dense in M, we have f = g.Thus φ is a uniform lifting of f .Now we turn to the general case.Since |M| < |V(Ξ)|, there exist isometries h : M → X and h : f [M] → X , where metric spaces X and X are in V(Ξ).Let e and e denote the canonical embeddings of X and X into their nonstandard hulls.Let ψ : M → * X be a uniform lifting of an isometry e • h : has a dense subspace of cardinality < κ.Therefore, there exists a uniform lifting ψ : See Henson and Moore [11] and Baratella and Ng [2] for some related results in the context of nonstandard hulls of * normed spaces.
Our goal is to extend Proposition 1.2 to a more general class of spaces that includes other types of nonstandard hulls, in particular, nonstandard hulls of topological vector spaces [10].Following Gordon [7], we consider topological spaces obtained as subspaces of quotients of internal sets by Π 0 1 (κ) equivalence relations.(A set is said to be An easy argument shows that is the base of some Hausdorff topology on M/R.(Note that all equivalence classes µ ∈ M/R are Π 0 1 (κ) and apply κ-saturation.)We call this topology the canonical topology on M/R.It follows from Proposition 2.4 below that this topology is generated by some family of pseudometrics and hence is Tychonoff.
We claim that the topology generated by d • c coincides with that induced by the canonical topology on If T ⊆ M is internal and µ ⊆ T , then by ω 1 -saturation there exists n ∈ N so that µ ⊆ b(x, n) ⊆ T , and it follows that for every n ∈ N, and the proof of our claim is complete.
Example 1.4 Let (K, τ ) be a compact Hausdorff space, K ∈ V(Ξ).Assume that there exists a base B of topology τ of cardinality < κ.It is well-known that * K is the disjoint union of monads µ K (x), x ∈ K ; let R = x∈K µ K (x) 2 be the corresponding equivalence relation.Let P denote the family of all pairs (U, V) ∈ B 2 such that cl U ⊆ V , where cl U is the closure of U in K .We have It is easily checked that the mapping e defined by e(x) = µ K (x) is a homeomorphism of K onto * K/R endowed with the canonical topology.
Example 1.5 Let (X, τ ) be a Tychonoff space, X ∈ V(Ξ).Assume that there exists a base B of topology τ of cardinality < κ.By [4, Theorem 3.5.2]there exists a compact Hausdorff space (K, τ ) containing (X, τ ) as a subspace and such that there exists a base of topology τ of cardinality < κ.By [4, Theorem 3.5.3]we may also assume that K ∈ V(Ξ).By the previous example Example 1.6 Let E ∈ V(Ξ) be a vector space (over F = R or C) and τ a Hausdorff topology on E such that (E, τ ) is a topological vector space.Assume that the filter of neighborhoods of 0 ∈ E has a base B 0 of cardinality < κ.Then The nonstandard hull of (E, τ ) is a topological vector space ( Ê, τ ) defined by the filter of neighborhoods of µ E (0) in Ê is generated by the sets st Ê( * U), U ∈ B 0 , see Henson [10].It is easily checked that the nonstandard hull topology τ coincides with that induced by the canonical topology on * E/R.(Apply Lemma 2.5 below.) For further examples see Luxemburg [12], Henson [9], Gordon [7,8], Mlček and Zlatoš [13], and Ziman and Zlatoš [14].
Let R (resp.R ) be a Π 0 1 (κ) equivalence relation on an internal set M (resp.M ) and f a mapping from M ⊆ M/R into M /R .We say that a mapping φ : Z → M is a weak uniform lifting of f if φ is an internal mapping with Z = dom φ ⊆ M such that µ ∩ Z = ∅ and φ[µ ∩ Z] ⊆ f (µ) for every µ ∈ M. A mapping φ is said to be a uniform lifting of f if φ is a weak uniform lifting of f with dom φ = M .We say that f is (weakly) uniformly liftable if there exists a (weak) uniform lifting of f .Every weakly uniformly liftable mapping is continuous, see Proposition 4.2 below.Gordon [7, Theorem 1.2] proved that if R is a Π 0 1 (κ) equivalence relation on an internal set M such that M/R is compact, then every continuous mapping f : M/R → fin( * C)/ ≈ is uniformly liftable.
We say that a topological space is κ-separable if it has a dense subspace of cardinality < κ.
The following are our main results: Theorem 1.7 Let R (resp.R ) be a Π 0 1 (κ) equivalence relation on internal set M (resp.M ) and M a κ-separable subspace of M/R.Then every continuous mapping f : M → M /R is weakly uniformly liftable.Moreover, a weak uniform lifting φ : H → M of f can be chosen so that H = dom φ is a hyperfinite subset of M .Theorem 1.8 Let R (resp.R ) be a Π 0 1 (κ) equivalence relation on internal set M (resp.M ) and M a subspace of M/R.Assume that at least one of the following conditions is satisfied: (4) M is separable and metrizable.
Then every continuous mapping f : M → M /R is uniformly liftable.Also we show that each of the following conditions (in the context of Theorem 1.8) is not sufficient for the existence of uniform liftings: (i) M is separable and f : M → * [0, 1]/ ≈ is continuous; (ii) M is separable and compact and f : M → M is a homeomorphism; see Example 4.6 below.

Alexander P Pyshchev
2 Generating families of * pseudometrics Let X be an arbitrary set and let U and V be the subsets of X 2 .Recall the following definitions: there exists y ∈ X so that (x, y) ∈ U and (y, z) ∈ V }, By ∆ X we denote the diagonal of X .
Lemma 2.1 Let X be a set and Proof This is an easy corollary of [4,Theorem 8.1.10].
Let R be an equivalence relation on internal set M and {d α : α ∈ A} a family of * pseudometrics on M .We say that R is generated by {d α : α ∈ A} and that Proof Let U be an internal set consisting of all internal sets U such that We may assume that {U α : α ∈ A} is closed under finite intersections.Clearly, R ⊆ α∈A U (2)  α .On the other hand, if (x, y) ∈ U (2)  α for all α ∈ A, then by κ-saturation α .By κ-saturation for every α ∈ A there exists β ∈ A such that U (2)  β ⊆ U α .Hence for every α ∈ A there exists γ ∈ A such that U (3)  γ ⊆ U (4)  γ ⊆ U α .Let f : A → A be such that U (3)  f (α) ⊆ U α for every α ∈ A.
Assume that an equivalence relation R on internal set M is generated by the family Lemma 4.3 Let X be a dense subspace of a Hausdorff space X , Y a subspace of a compact Hausdorff space Ỹ , and f : X → Y a continuous mapping.Then there exists an extension f : X → Ỹ of f which is continuous at every point of X .
Proof (It is convenient to use nonstandard analysis at every step.)Let F be the closure of f in X × Ỹ .Since X is dense in X and Ỹ is compact, dom F = X .Since f is continuous, F ∩ (X × Ỹ) = f .Let f : X → Ỹ be any mapping such that f ⊆ F ; then f is continuous at every point of X .
Lemma 4.4 Let X be a nonempty closed subspace of metrizable space X .Then there exists a mapping r : X → X such that r(x) = x for every x ∈ X and which is continuous at every point of X .
Proof Let d be any compatible metric on X .For every y ∈ X choose r(y) ∈ X so that d(y, r(y)) ≤ 2d(y, X).
Lemma 4.5 Let X be a Lindelöf subspace of a Tychonoff space X and Y a subspace of a compact Hausdorff space Ỹ .Assume that at least one of the spaces X and Y is separable and metizable.Then for every continuous mapping f : X → Y there exists an extension f : X → Ỹ which is continuous at every point of X .
Proof In both cases there exist continuous mappings g : X → Z and h : Z → Y such that Z ⊆ [0, 1] N and h • g = f .(Note that every separable metrizable space embeds into [0, 1] N .)Since X is Lindelöf, by [1,Corollary 16] for every n ∈ N there exists an extension gn : X → R of g n = pr n • g which is continuous at every point of X .We may assume that gn : X → [0, 1].Let g : X → [0, 1] N be such that gn = pr n • g for every n ∈ N; it is easily checked that g is an extension of g which is continuous at every point of X .
Let K be the closure of Z in [0, 1] N .By Lemma 4.4 there exists a mapping r : [0, 1] N → K such that r(t) = t for every t ∈ K and which is continuous at every point of K .By Lemma 4.3 we obtain an extension h : K → Ỹ of h which is continuous at every point of Z .It is clear that f = h • r • g is an extension of f which is continuous at every point of X .
Proof of Theorem 1.8 Case (1): By Theorem 1.7 there exists a weak uniform lifting ψ : H → M of f such that H is a hyperfinite subset of M .By Proposition 2.3 there exists a * metric d on M with R = d −1 (µ R (0)).Since H is hyperfinite, by transfer there

Proposition 1 . 2
Let f : M → H be a continuous mapping from a subspace M of the nonstandard hull H of a * metric space (M, d) around c ∈ M into the nonstandard hull H of a * metric space (M , d ) around c ∈ M .If M has a dense subspace of cardinality < κ and |M| < |V(Ξ)|, then f is uniformly liftable.
generated by the family {d α : α ∈ A}.Proposition 2.3 Every Π 0 1 equivalence relation on internal set M is generated by some * metric d on M .Proof Applying Proposition 2.2 with κ = ω 1 , we obtain a generating family {d n : n ∈ N} for R. We may assume that all * pseudometrics d n take values in * [0, 1] (e.g., we may consider a * pseudometric dn (x, y) = min{d n (x, y), 1} instead of d n ).Let (d ν ) 0≤ν≤H be an internal extension of (d n ) n∈N which also consists of * pseudometrics on M with values in * [0, 1].We may assume that d H is a {0, 1}-valued * metric on M .Put d(x, y) = H ν=0 d ν (x, y)/2 ν+1 .

Proposition 2 . 4
Let R be a Π 0 1 (κ) equivalence relation on internal set M and {d α : α ∈ A} a generating family of * pseudometrics for R with |A| < κ.Then the topology on M/R generated by pseudometrics d • α , α ∈ A, coincides with the canonical topology.