Monatsh Math (2018) 186:337–344 https://doi.org/10.1007/s00605-018-1157-z On isometric embeddings and continuous maps onto the irrationals 1 1 Elzbieta ˙ Pol · Roman Pol Received: 12 January 2017 / Accepted: 9 January 2018 / Published online: 17 January 2018 © The Author(s) 2018. This article is an open access publication Abstract Let f : E → F be a continuous map of a complete separable metric space E onto the irrationals. We shall show that if a complete separable metric space M contains isometric copies of every closed relatively discrete set in E, then M contains −1 also an isometric copy of some ﬁber f (y). We shall show also that if all ﬁbers of f have positive dimension, then the collection of closed zero-dimensional sets in E is non-analytic in the Wijsman hyperspace of E. These results, based on a classical Hurewicz’s theorem, reﬁne some results from Pol and Pol (Isr J Math 209:187–197, 2015) and answer a question in Banakh et al. (in: Pearl (ed) Open problems in topology II. Elsevier, Amsterdam, 2007). Keywords Isometric embeddings · Effros Borel spaces · Zero-dimensional sets Mathematics Subject Classiﬁcation Primary 54E40 · 54F45 · 54H05 1 Introduction In [13] we proved that each complete separable metric space containing isometric copies of every countable complete metric space contains isometric copies of every separable metric space. Communicated by A. Constantin. Elzbieta ˙ Pol E.Pol@mimuw.edu.pl Roman Pol R.Pol@mimuw.edu.pl University of Warsaw, Warsaw, Poland 123 338 E. Pol, R. Pol We shall reﬁne this result to the following effect. Theorem 1.1 Let f : E → F be a continuous map of a complete separable metric space onto a non-σ -compact metric space. Then there exists a relatively discrete set S in E such that, for any complete separable metric space M containing isometric −1 copies of every subset of S closed in E, some ﬁber f (y) embeds isometrically in M. The result from [13] follows from this theorem, if we consider the restriction map f : C [0, 1]→ C [ , 1] (recall that by the Banach–Mazur theorem, cf. [8, Theorem 5.17], the space (C (I ), d ) of all real-valued continuous functions on the interval sup I =[0, 1], equipped with the metric d ( f, g) = sup{| f (t ) − g(t ) |: t ∈ I },is sup isometrically universal for all separable metric spaces). Also, as in [13], one can replace in this theorem isometries by uniform homeomor- phisms. The proofs will go along the same lines as in [13], and an essential part of the reasonings can be taken directly from [13], cf. Sect. 4. However, a classical Hurewicz’s theorem on non-analyticity of the set of compact subsets of the rationals is applied in a different way than in [13]. We shall prove a result based on the Hurewicz theorem in a slightly more general form than needed for Theorem 1.1 in Sect. 3, to establish a link with some questions concerning the dimension, discussed in Sect. 5. 2 Preliminaries 2.1 The Effros Borel spaces Our terminology related to the descriptive set theory follows [7,9]. An analytic space is a metrizable continuous image of the irrationals. Given an analytic space E, we denote by F (E ) the space of closed subsets of E and B —the Effros Borel structure in F (E ),isthe σ -algebra in F (E ) generated F (E ) by the sets {A ∈ F (E ) : A ∩ U =∅}, where U is open in E. We shall say that A ⊂ F (E ) is a Souslin set in the Effros Borel space (F (E ), B ) F (E ) if A is a result of the Souslin operation on sets from B . F (E ) If X is a compact metrizable space, we shall consider the hyperspace F (X ) with the Vietoris topology and then B coincides with the σ -algebra of Borel sets in the F (X ) compact metrizable space F (X ). If X is a compact metrizable extension of an analytic set E ⊂ X,the map A → A (the closure is taken in X)from F (E ) to F (X ) is a Borel isomorphism, with respect to the Effros Borel structures, onto the analytic subspace {A : A ∈ F (E )} of the hyperspace F (X ) and hence Souslin sets in F (E ) are mapped onto analytic sets in F (X ),cf. [7, Section 2]. In particular, if E ⊂ G ⊂ X and G is analytic, the collection of closures of elements of F (E ) in G is a Souslin set in F (G). 2.2 The Hurewicz theorem Let I =[0, 1] and let Q be the set of rationals in I . 123 On isometric embeddings and continuous maps onto the irrationals 339 The classical Hurewicz theorem asserts that any Souslin set in F (I ) containing all compact subsets of Q, contains an element intersecting I \Q. We shall derive from this theorem the following observation, which we shall use in the next section. Let us arrange points of Q into a sequence q , q ,... (without repetitions), let 1 2 D = q , : n = 1, 2,..., m ≥ n , L = (I \Q) ×{0}, (2.1) let T = L ∪ D (2.2) be the subspace of the plane (notice that D is relatively discrete in T ), and let D ={A ⊂ D : A is closed in T }. (2.3) Lemma 2.2.1 For any Souslin set E in F (T ) containing D, some element of E inter- sects L. Proof For A ∈ F (T ), A will denote the closure in the plane. As was recalled in 2.1, the set {A : A ∈ E} is analytic in F (T ) (notice that T = (I ×{0}) ∪ D), hence the set {(K , A) ∈ F (I ) × F (T ) : A ∈ E and K ×{0}⊂ A} is analytic in the product of the hyperspaces, and so is its projection onto F (I ), E ={K ∈ F (I ) : K ×{0}⊂ A for some A ∈ E}. (2.4) If K ⊂ Q is compact, A = D ∩ (K × I ) is closed in T and K ×{0}⊂ A, hence K ∈ E ,cf. (2.4). By the Hurewicz theorem, there is A ∈ E such that A intersects L, cf. (2.1) and (2.4), and since A is closed in T , A intersects L. 2.3 A remark on continuous maps onto the irrationals We shall need the following observation. This is close to some well-known results, but for readers convenience, we shall provide a brief justiﬁcation. Lemma 2.3.1 Let f : E → F be a continuous map of an analytic space onto a non- σ -compact metrizable space. There is a closed copy of the irrationals P in F and continuous maps g : P → E such that, for each t ∈ P, {g (t ) : n = 1, 2,...} is a n n −1 dense subset of f (t ). Proof Let p : N → E be a continuous surjection of the irrationals onto the analytic space E. Then u = f ◦ p : N → F is a continuous surjection onto a non-σ -compact metrizable space and one can ﬁnd a closed copy of the irrationals P in F such that the −1 −1 restriction map u | u (P) : u (P) → P is open, cf. [14, proof of Theorem 3.1]. 123 340 E. Pol, R. Pol By a selection theorem of Michael [10], one can deﬁne a sequence of continuous −1 −1 selections w : P → u (P) for the lower-semicontinuous multifunction t → u (t ) −1 such that, for each t ∈ P,the set {w (t ) : n = 1, 2,...} is dense in u (t ). −1 Then the functions g = p ◦ w : P → f (P) satisfy the assertion. n n 3 An application of the Hurewicz theorem The following proposition strengthens a known fact that, for the irrationals N ,any N N Souslin set in F (N ) containing all countable closed sets in N , contains also a non- σ -compact set (this is stated in [9, Exercises 27.8, 27.9]; to derive this fact from the N N N proposition, notice that N is homeomorphic to N × N and consider the projection N N N N × N → N ). The setting is a bit more general than needed for Theorem 1.1, but it is useful to establish connections with some topics in the dimension theory, discussed in Sect. 5. Proposition 3.1 Let f : E → F be a continuous map of an analytic space onto a non-σ -compact metrizable space. Then there exists a relatively discrete set S in E such that for any Souslin set A in F (E ) containing all subsets of S closed in E, there −1 are A ∈ A and y ∈ F with f (y) ⊂ A. Proof Let P be a closed copy of the irrationals in F and g : P → E continuous maps described in Lemma 2.3.1, and let T = L ∪ D be the subspace of the plane deﬁned in (2.1) and (2.2). Since T is a zero-dimensional G -subset of the plane, there is a homeomorphic embedding h : T → P, h(T ) closed in P. (3.1) Let us arrange points of D into a sequence without repetitions D ={d , d ,...} and c = h(d ). (3.2) 1 2 n n We shall check that, cf. (3.2), S ={g (c ) : n = 1, 2,..., m ≤ n}⊂ E (3.3) m n satisﬁes the assertion of the proposition. −1 Since g (c ) ∈ f (c ), f (S) = h(D) is relatively discrete and S intersects each m n n ﬁber of f in at most ﬁnite set, cf. (3.3). Therefore S is relatively discrete. Let, for X ∈ F (T ), −1 −1 ϕ(X ) = f (h(X ∩ L)) ∪ (S ∩ f (h(X ∩ D))). (3.4) −1 Since all accumulation points of S in E are in f (h(L)) and h(X ) is closed in F,cf. (3.1), we have ϕ(X ) ∈ F (E ). 123 On isometric embeddings and continuous maps onto the irrationals 341 We shall check that ϕ : F (T ) → F (E ) is Borel, (3.5) with respect to the Effros Borel structure. To that end, let us ﬁx an open set U in E, and let U ={X ∈ F (T ) : ϕ(X ) ∩ U =∅}. (3.6) Let X ∈ U.Iffor some m ≤ n, d ∈ X and g (c ) ∈ U,cf. (3.2), (3.3), (3.4), the n m n element {Y ∈ F (T ) : d ∈ Y } of B contains X and is contained in U. n F (T ) −1 Let a ∈ X ∩ L and f (h(a)) ∩ U =∅. Since the points g (h(a)) are dense in −1 f (h(a)), there is m such that g (h(a)) ∈ U.Let V be a neighbourhood of h(a) in F such that g (V ) ⊂ U, and let us pick a rectangle J = (r, s) ×[0, ) disjoint from {d ,..., d } with r, s ∈ Q, containing a, such that h(J ∩ T ) ⊂ V.If Y ∈ F (T ) hits 1 m −1 J , there is either b ∈ Y ∩ L with h(b) ∈ V and then f (h(b)) ⊂ ϕ(Y ) intersects U, or there is d ∈ Y ∩ J with n > m and then, cf. (3.3), (3.4), g (c ) ∈ ϕ(Y ) ∩ U. n m n Therefore the element {Y ∈ F (T ) : Y ∩ J =∅} of B contains X and is F (T ) contained in U. We demonstrated that U is a countable union of elements of B , hence belongs F (T ) to the Effros Borel structure of F (T ). Having checked (3.5), let us consider the set S ={A ⊂ S : A ∈ F (E )} (3.7) and let S ⊂ A, A is Souslin in (F (E ), B ). (3.8) F (E ) By (3.5), −1 E = ϕ (A) is Souslin in (F (T ), B ). (3.9) F (T ) If X ⊂ D is closed in T , h(X ) is closed in F and ϕ(X ) is closed in E,cf. (3.4), hence ϕ(X ) ∈ S,cf. (3.7). Therefore, by (3.8), for the set D deﬁned in (2.3), we have D ⊂ E and Lemma 2.2.1 provides X ∈ E and a point a ∈ X ∩ L.By(3.4) and (3.9) we get −1 A = ϕ(X ) ∈ A and f (h(a)) ⊂ A. 4 Proof of Theorem 1.1 We shall recall brieﬂy some reasonings from [13] to derive this theorem from Propo- sition 3.1. Given f : E → F as in this theorem, let us pick S satisfying the assertion of Proposition 3.1. 123 342 E. Pol, R. Pol Let e be the complete metric on E and let (M, d) be any complete separable metric space, containing isometric copies of every subset of S closed in E.Let H ={T ∈ F (E × M ) : for every (x , y ), (x , y ) ∈ T , e(x , x ) = d(y , y )}. 1 1 2 2 1 2 1 2 One checks, cf. [13, p. 193], that H is in B and the map T → π(T ) associating F (E ×M ) to T ∈ H its projection onto E is a Borel map π : H → F (E ). Therefore A = π(H) is a Souslin set in (F (E ), B ).If X ⊂ S is closed in E, F (E ) there is an isometry f : X → f (X ) ⊂ M and the graph of f is an element of H. It follows that the Souslin set A contains all subsets of S closed in E, and by the −1 choice of S,some A ∈ A contains a ﬁber f (y). Now, A = π(T ) and T is the graph of an isometry that embeds A in M. In effect, −1 f (y) embeds isometrically in M. 5 The collections of zero-dimensional sets in Effros Borel spaces Our terminology concerning the dimension theory follows [15]. Given an analytic space, we shall write F (E ) ={A ∈ F (E ) : dimA = 0}. (5.1) We shall derive from Proposition 3.1 the following result. Proposition 5.1 Let E be an analytic space that admits a continuous map f : E → F −1 onto a non-σ -compact metrizable space such that all ﬁbers f (y) have positive dimension. Then for any analytic extension G of E with dim(G\E ) ≤ 0,the set F (G) is not Souslin in the Effros Borel space (F (G), B ). 0 F (G) Proof By Proposition 3.1, there is a relatively discrete set S in E such that for any Souslin set A in F (E ) containing S ={A ∈ F (E ) :∅ = A ⊂ S}, some element of the set A contains a ﬁber of f and hence has positive dimension. Now, consider an analytic extension G of E with dim(G\E ) ≤ 0 and, aiming at a contradiction assume that F (G) is Souslin in F (G). As was recalled in Sect. 2.1, the map A → A from F (E ) to F (G) is Borel, and hence we would get that the set A ={A ∈ F (E ) : dimA ≤ 0} is Souslin in F (E ). If A ∈ S, then A is a relatively discrete closed set in E, and hence A\A is a closed subset of G contained in G\E. This implies that dimA = 0, i.e., S ⊂ A. However, all members of A are zero-dimensional, which contradicts properties of S. In particular, if P is the set of irrationals in I =[0, 1], F (P × I ) is not Souslin in F (P × I ) (5.2) (this rectiﬁes a remark in [5, §3.A]). Banakh et al. [1, Question 9.12], asked about the Borel type of the collection F (E ) in the space CL(E ) = F (E )\{∅}, when E is a completely metrizable separable space, 123 On isometric embeddings and continuous maps onto the irrationals 343 and CL(E ) is considered with the Wijsman topology τ , determined by some metric d generating the topology of E (i.e., τ is the weakest topology making all functionals A → dist(z, A), z ∈ E, continuous), cf. [2,4]. The Wijsman hyperspace (CL(E ), τ ) is completely metrizable, separable, cf. [6], and the Borel sets with respect to τ coincide with the members of the Effros Borel structure in CL(E ). Therefore, F (P × I ) is not a Borel (or even Souslin) set in CL (P × I ). (5.3) One can check that its complement F (P × I )\F (P × I ) is Souslin. Let us consider, 2 2 2 2 however, the subspace I \Q of the square, Q = I \P. Since (I \Q )\(P×I ) = Q×P 2 2 2 2 is zero-dimensional, also F (I \Q ) is not Souslin in F (I \Q ), by Proposition 5.1. 2 2 2 2 But it is not clear to us whether F (I \Q )\F (I \Q ) is Souslin. In fact, we do not know an answer to the following general question. Question 5.2 Does there exist an analytic space E such that F (E )\F (E ) is not Souslin in the Effros Borel structure? This question is related to the following problem, asked in [11], where countable- dimensional spaces are countable unions of zero-dimensional spaces. Problem 5.3 Is the collection C of all countable-dimensional compact sets in the N N Hilbert cube I coanalytic in the hyperspace F (I ) equipped with the Vietoris topol- ogy? To see the link between these two questions, let us consider a Borel set E ⊂ I such that I \E is countable-dimensional and all countable-dimensional subsets of E are at most zero-dimensional, cf. [12]. We shall assume in addition that E is disjoint from the set consisting of points in I with all but ﬁnitely many coordinates zero. N N N By [3, Ch.V, §5], there is a homeomorphism h : I \ → R × R (where R is N N N the real line), let p : R × R → R be the projection and let f = p ◦ h | E : E → R . Then f is a continuous surjection whose all ﬁbers are uncountable-dimensional. Therefore, by Corollary 5.1, F (E ) is not Souslin in the Effros Borel space. We do not know if the set E = F (E )\F (E ) is Souslin. Let us show, however, that if this is the case, C in Problem 5.3 is coanalytic. Suppose that E is Souslin in (F (E ), B ). Then, as was noticed in Sect. 2.1, F (E ) the collection E ={A : A ∈ E} of the closures in I is analytic in the hyperspace N N F (I ).Now, K ∈ F (I )\C if and only if K ∩ E is uncountable-dimensional, which is equivalent to K ∩ E ∈ / F (E ). Therefore, F (I )\C is the projection of the analytic N N set {(K , L) ∈ F (I ) × F (I ) : L ⊂ K and L ∈ E }, hence it is analytic. Added in the revision Concerning Question 5.2, Debs and Saint Raymond gave in a recent paper “The descriptive complexity of the set of all closed zero-dimensional subsets of a Polish space” a subtle construction of a G -set E in I such that F (E ) δ 0 is not even a C-set in F (E ) (in particular, F (E )\F (E ) is not Souslin). The question 2 2 concerning F (I \Q ) remains open. 123 344 E. Pol, R. Pol Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. References 1. Banakh, T., Cauty, R., Zarichnyi, M.: Open problems in inﬁnite-dimensional topology. In: Pearl, E. (ed.) Open Problems in Topology II. Elsevier, Amsterdam (2007) 2. Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic, Dordrecht (1993) 3. Bessaga, C., Pełczynski, ´ A.: Selected Topics in Inﬁnite-Dimensional Topology, PWN-Polish Scientiﬁc Publishers, Warsaw, 1975, Monograﬁe Matematyczne, tom 58 4. Chaber, J., Pol, R.: Note on the Wijsman hyperspaces of completely metrizable spaces. Bull. U.M.I (8) 5–B, 827–832 (2002) 5. Clemens, J.D., Gao, S., Kechris, A.S.: Polish metric spaces: their classiﬁcation and isometry groups. Bull. Symb. Logic 7(3), 361–375 (2001) 6. Costantini, C.: Every Wijsman topology relative to a Polish space is Polish. Proc. Am. Math. Soc. 123, 3393–3396 (1998) 7. Dellacherie, C.: Un Cours Sur Les Ensembles Analytiques. In: Rogers, C.A. (ed.) Analytic Sets. Academic Press, London (1980) 8. Fabian, M., Habala, P., Hajék, P., Montesinos Santalucía, V., Pelant, J., Zizler, V.: Functional Analysis and Inﬁnite-Dimensional Geometry. Springer, New York (2001) 9. Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995) 10. Michael, E.: Selected selection theorems. Am. Math. Mon. 63, 233–238 (1956) 11. Pol, R.: On classiﬁcation of weakly inﬁnite-dimensional compacta. Fundam. Math. 116, 169–188 (1983) 12. Pol, R.: Countable dimensional universal sets. Trans. Am. Math. Soc. 297, 255–268 (1986) 13. Pol, E., Pol, R.: Note on isometric universality and dimension. Isr. J. Math. 209, 187–197 (2015) 14. Pol, R., Zakrzewski, P.: On Borel mappings and sigma-ideals generated by closed sets. Adv. Math. 231, 651–663 (2012) 15. van Mill, J.: The Inﬁnite Dimensional Topology of Function Spaces. North-Holland, Amsterdam (2001)
Monatshefte f�r Mathematik – Springer Journals
Published: Jan 17, 2018
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