Topology Proceedings

Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124

THE STRONG UNIVERSALITY OF CERTAIN STARLIKE SETS AND APPLICATIONS

Tadeusz Dobrowolski

Abstract Following Banakh’s idea [Ba], a version of strong universality in certain starlike sets is verified. Ap- plications to an embedding problem for such starlike sets and sigma-compact absorbers follow. This paper aims at revisiting Banakh’s article [Ba]. In [Ba], extending the ideas from [DM1], [Do], [BD] [BC1], and from some other papers, T. Banakh made an important contribution to the topological classification of incomplete convex sets. Let us recall that the main tool for obtaining such a classification is the so-called absorbing set method (see e.g. [BP], [vM], and [BRZ]). This method requires to verify: (1) the so-called strong universality property of X, the space in question, for a class of spaces C, (2) the Zσ-property of X (both defined is Section 2), and (3) that X can be expressed as a countable union of elements of C that are closed in X. An intriguing, very concrete question is whether a σ-compact convex subset C of `2 is strongly universal for the class of compacta embeddable in C. An answer to this question is also unknown if, additionally, C has the Zσ- property, cf. [DM2, Question 5.6]. In general, as shown in Section 2 there exists an example of a convex subset of `2 that has the Zσ-property and is not strongly universal for the class of its closed subsets. The central role in Banakh’s approach [Ba] is played by certain special homeomorphisms, which, in our paper, are called Mathematics Subject Classification: 57N17, 57N20, 46A55, 52A07. Key words: Convex set, almost internal point, starlike set, strong universality, absorber. 80 Tadeusz Dobrowolski almost cylindrical. Employing some properties of such homeomorphisms, Banakh verifies a certain version of the strong universality property for convex sets with almost internal points. Inspecting his reasoning, we re-examine the use of cylindrical homeomorphisms to extend his result to some sets X with C ⊆ X ⊆ C¯ for some convex set C with an almost internal point. All applications that are listed in [Ba] can be carried over to such sets; we decided not to include them in this text (because this would be a formality only) with one exception related to embeddings of such sets into `2. We provide one more application concerning σ-compact absorbing sets that behave strangely with respect to the Cartesian product and do not admit any reason- able algebraic structure. These absorbing sets are versions of examples given in [BC2] (also cf. [Za]).

1. Mappings into Almost Convex Bodies

∞ n n For x =(xi) ∈ R ,weletpn(x)=(x1,...,xn) ∈ R p (x)= n (xn+1,xn+2,...), and πi(x)=xi. We identify p (x) with ∞ (0,...,0,xn+1,xn+2,...) ∈ R and, making an obvious identification of Rn with a subspace of ∞ ∞ ∞ R , we view pn(x)as(x1,...,xn, 0,...) ∈ R . The space R will be considered with the metric induced by the F -norm kxk = −i ∞ max{2 min{|xi|, 1}| i ≥ 1}. For two maps f,g : X → R ,we let kf − gk = sup{kf(x) − g(x)k|x ∈ X}. We say that a homeomorphism h ∈ H(R∞)isn-cylindrical if there exists a homeomorphism h¯ ∈ H(Rn) such that h(x)= ¯ n ∞ (h(pn(x)),p (x)), x ∈ R . We say that h is almost n-cylindrical if, for some continuous function α : Rn → (0, ∞), ¯ n ∞ h(x)=(h(pn(x)),α(pn(x))p (x)), x ∈ R . The set of all n-cylindrical homeomorphisms and almost n-cylindrical home- ∞ a ∞ omorphisms is denoted by Hn(R ) and Hn(R ), respectively. a ∞ ∞ a ∞ ∞ a ∞ We let H (R )=Sn=1 Hn(R ). Notice that Hn(R ), Hn(R ), ∞ ∞ a ∞ Sn=1 Hn(R ), and H (R ) are groups. Such n-cylindrical and THE STRONG UNIVERSALITY OF CERTAIN STARLIKE ... 81 almost n-cylindrical homeomorphisms have been frequently used in infinite-dimensional topology, especially in order to construct certain smooth maps on a topological vector space E. Then, R∞ is replaced by E, and E is required to admit a splitting n E = En ⊕ E , where En is finite-dimensional. ∞ ∞ Remark 1. Let E be a linear space with Rf ⊆ E ⊆ R , ∞ ∞ n n where Rf = Sn=1 R . Since pn(x),p (x)=x − pn(x) ∈ E for every x ∈ E, we have h(E)=E for every homeomorphism ∞ a ∞ h ∈ Hn(R ) ∪ Hn(R ). Actually, the above is true if E is ∞ ∞ replaced by X ⊆ R with RX + Rf = X. We say that a convex set C ⊆ R∞ is an almost convex ∞ ∞ body relative Rf ⊂ R if,

n (a) 0 ∈ intRn (C ∩ R ) for every n, and

∞ (b) Rf ∩ C is dense in C. Recall that a set X ⊆ R∞ is star-shaped with respect to x ∈ X if, for every y ∈ X, the segment [y,x]={ty+(1−t)x| 0 ≤ t ≤ 1} is contained in X. The kernel, ker(X), of X is the set of all x ∈ X with respect to which X is star-shaped; ker(X) is always a convex set. Recall that by the radial interior of a set X that is star-shaped with respect to 0 we mean the set rint(X)=[0, 1)X.

The following is a counterpart of [Ba, Lemma 2]. Lemma 1. Let C ⊆ R∞ be an almost convex body and K be a compactum with K ⊆ C¯. For every h ∈ Ha(R∞), C absorbs h(K ∩ C).IfX is such that ker(X)=C, then X absorbs h(K ∩ X).

∞ Proof. If h ∈ Hn(R ) then, using the fact that 0 ∈ n intRn (C ∩ R ) and the compactness of K, there exists β ≥ 1 such that n pn(h(K)) ∪ pn(K) ⊂±β(C ∩ R ). 82 Tadeusz Dobrowolski Now, for x ∈ K ∩ X,wehave

h(x)=pn(h(x)) + (x − pn(x)) ∈ β(C ∩ Rn)+X + β(C ∩ Rn) ⊆ (2β +1)X. The last inclusion follows from the fact that X is starlike with respect to any point of C. A similar argument can be used to show that X absorbs g(K∩X) a ∞ n for g ∈ H (R ) of the form g(x)=(pn(x),α(pn(x))p (x)). It is not diﬃcult to show that the properties in question are preserved by the composition of homeomorphisms. Now, since every h ∈ Ha(R∞) is a composition of homeomorphisms considered above, the assertion of Lemma 1 follows.

Lemma 2. Let A ⊂ R∞ and B ⊂ Rk be compacta, and f : B → Rk be a map such that [B ∪ f(B)] ∩ A = ∅. Then, for ε>0, ∞ there exist n ≥ k and h ∈ Hn(R ) such that

(i) kh − id k < kf − idB k + ε,

(ii) h|A =idA, (iii) kh|B − fk <ε. Proof. Inspect [DM1, Lemma 1].

In the proof below we use the fact that the compacta are so- called Z-sets in R∞. For information on Z-sets and locally homotopy negligible sets see [vM], [BRZ], and [To]. Here, we only recall that a closed subset A of an ANR space X is a Z-set if every map of a compactum into X can be arbitrarily closely approximated by maps whose ranges miss A; X is a Zσ- space (or, has Zσ-property) if it can be expressed as a countable union of Z-sets. Lemma 3. Let A and B be disjoint compacta in R∞. Then, given a map f : B → R∞ and ε>0 there exists an h ∈ Ha(R∞) such that THE STRONG UNIVERSALITY OF CERTAIN STARLIKE ... 83

(i) kh − id k < kf − idB k + ε,

(ii) h|A =idA, (iii) kh|B − fk <ε.

Proof. Though the proof can easily be obtained by inspecting the proof of (a particular case of) [DM1, Proposition] we have decided to provide the details for it. First, using the fact that A is a Z-set, we can approximate f by a map whose range is disjoint from A. So, we can assume that f(B) ∩ A = ∅. Extend f to a map f¯ : R∞ → R∞.For ε>0 there exists δ>0 such that

[b ∈ B and kb − b0k <δ] ⇒kf¯(b) − f¯(b0)k <ε/4.

−k Choose k so large that 2 <δand pk(B) ∩ pk(A)=∅. Let 0 ¯ 0 0 0 k0 B = pk(B) and approximate f|B by f : B → R such that k0 ≥ k, kf¯|B0 − f 0k <ε/4 and f 0(B0) ∩ A = ∅. Increasing k0, if necessary, we furthermore approximate f 0 by an embedding v (e.g., having the form f 0 +w with kwk suﬃciently small) so that

0 0 kv − idB0 k < kf − id k + δ and kv − f k <δ, and v(B) ∩ A = ∅. By Lemma 2, we can extend v to a homeo- 0 ∞ morphism h ∈ Hn(R ) that satisﬁes

0 0 0 0 0 0 kh |B −f k<ε/4, kh −id k

∞ We need a “pseudohomotopy” u =(ut):R ×[0, 1)∪B×{1}→ ∞ R such that u1 = pk|B and, for t<1,

a ∞ ut ∈ H (R ), kut − id k <δ, and ut|A =idA .

Having such a u, it is easy to see that, for t suitably close to 1, t< 0 1, the map h = h ◦ ut may serve as a required homeomorphism that satisfies (i)–(iii). 84 Tadeusz Dobrowolski To construct u, let first λ : Rk → [0, 1] be a continuous function with λ|pk (B)=0andλ|pk(A) = 1. Pick any homotopy (αt)0≤t<1 of [0, 1] such that 0 <αt ≤ 1 and αt(1) = 1 for all t, α0 ≡ 1, and limt→1 αt(s) = 0 for each s<1. Finally, for t<1, define k ut(x)=(pk(x),αt(λ(pk(x)))p (x)).

For a set A ⊂ R∞ and a map f : A → R∞, we write f ∈ Ha(A) and say that f is locally almost cylindrical if, every point x ∈ A admits a neighborhood U in A such that f|U = h|U for some h ∈ Ha(R∞). If A is locally compact then f ∈ Ha(A) if, for every compactum K ⊆ A, f|K = h|K for some h ∈ Ha(R∞).

Lemma 4. Given a map f : K → R∞, where K ⊂ R∞ is a compactum, and a continuous function ε : K → [0, 1], there exists a map g : K → R∞ such that (i) kg(x) − f(x)k≤ε(x), x ∈ K; (ii) g|A ∈ Ha(A), where A = K \ ε−1(0).

Proof. We only need to slightly modify a procedure from the proof of [DM1, Theorem] (see [Ba]). Choose the tower An = −n ∞ {x ∈ K| ε(x) ≥ 2 }; we have Sn=1 An = A. Construct a a ∞ sequence {gn}⊂H (R ) such that

−n kgn − gn−1k < 2 ,gn|An−1 = gn−1|An−1, and −n−2 kgn(x) − f(x)k < 2 for x ∈ K \ int(An). The last condition (which in [DM1] was required only if ε(x)= ¯ 0) guarantees that, for f(x) = lim gn(x), x ∈ K, we have that ¯ −(n−1)−2 −n−1 kf(x) − f(x)k = kgn(x) − f(x)k≤2 =2 <ε(x) ¯ for x ∈ An \ An−1, and obviously that f(x)=f(x) when ε(x)= 0. THE STRONG UNIVERSALITY OF CERTAIN STARLIKE ... 85 a ∞ To construct the sequence {gn}, let g1 ∈ H (R ) be such k | − k 1 that g1 K f < 8 . Having found g1,...,gn−1, apply Lemma 3 to the quadruple [A = gn−1(An−1), B = gn−1(K \ int(An)), −1 ∞ −n−2 f ◦ gn−1 : B → R , ε =2 ]. Since, for y = gn−1(x) ∈ −1 −n−1 B, we have kf ◦ gn−1(y) − yk≤kf(x) − gn−1(x)k≤2 , a ∞ by Lemma 3, there exists h ∈ H (R ) such that h|A =idA, −1 −n−2 −n−1 −n−2 −n kh − id k < kf ◦ gn−1|B − idB k +2 < 2 +2 < 2 , −1 −n−2 and kh|B − f ◦ gn−1|Bk < 2 . We let gn = h ◦ gn−1 ∈ Ha(R∞).

Lemma 5. Let C be an almost convex body in R∞. For a set X ⊆ C¯ with ker(X)=C, let K ⊆ X be compact and B be a closed subset of K.Letf : K → X be a map such that f|B is injective and f(K \ B) ∩ f(B)=∅, and let ε>0. Then there exists an embedding f¯ : K → X such that

(i) f¯|B = f|B;

(ii) kf¯− fk <ε;

(iii) f¯((K ∩ C) \ B) ⊆ rint(C);

(iv) f¯(K \ B) ⊆ rint(X);

(v) f¯|K\B ∈ Ha(K\B); in particular, f¯−1(E)\B =(K∩E)\B ∞ ∞ for every linear space E such that Rf ⊆ E ⊆ R .

Proof. The proof follows the arguments of [Ba, Lemma 7 and 8]. Namely, a sequence of compacta

∅=K−2 =K−1 =K0 ⊂ K1 ⊂ K2 ⊂···,Kn ⊂ int(Kn+1), and ∞ Sn=−2 Kn = K \ B, a sequence of natural numbers k0

(1n) gn(Kn) ⊂ rint(X) and gn(Kn ∩ C) ⊂ rint(C);

(2n) gn(x)=gn−1(x) for x ∈ Kn−2 ∪ (K \ int(Kn+1));

(3n) gn(x) ∈ [pkn+1 (gn−1(x)),gn−1(x)] that is πi(gn(x)) = πi(gn−1(x)) for all i ≤ kn+1 and πi(gn(x)) ∈ [0, 1]πi(gn−1(x)) for i>kn+1. ¯ Having done this, let f(x) = lim gn(x), x ∈ K. Note that conditions (i)-(v), except (ii), follow (for the “particular” part of (v) apply the statement of Remark 1); (ii) will be checked below. The inductive procedure of ﬁnding the sequence {gn} starts with the construction of g0 that will determine sequences (Kn) and (kn). We will additionally require that g0 satisﬁes

(4) g0|B = f|B;

(5) kg0 − fk <ε/2;

kn kn (6n) pkn ◦ g0(Kn \ Kn−3) ⊂ intRkn (C ∩ R ) = rint(C ∩ R );

(7n) pkn ◦ g0(Kn) ∩ pkn ◦ g0(K \ int(Kn+1)) = ∅.

In order to fulfill (6n) and (7n), we first replace f by (a close to 0 f) map f that satisfies these conditions for some sequence (Kn), and next we find g0 by an application of Lemma 4. We use the ∞ fact that the complement of rint(C) ∩ (Rf \ f(B)) is a locally homotopy negligible set in X \ f(B) to find a map f 0 such that 0 ∞ 0 0 0 f (K \ B) ⊂ rint(C) ∩ Rf , f |B = f|B and f (K \ B) ∩ f (B)= ∅. Moreover, we can easily additionally achieve that, locally, f 0|K \ B takes values in a certain Rn ⊂ R∞, that is,

(∗) ∀ (x ∈ K \ B) ∃ (n and a neighborhood Ux of x) 0 n [f (Ux) ⊂ intRn (C ∩ R )]. Agree that the original f has the above properties of f 0. De- ﬁne inductively (Kn) and (kn) as follows. Let K0 = ∅. Choose −k0 a number k0 ∈ N with 2 <ε/2 (this choice yields (ii) by ¯ an application of (5) because pk2 (f(x)) = pk2 (g0(x)) implies THE STRONG UNIVERSALITY OF CERTAIN STARLIKE ... 87

¯ −k2 −k0 kf(x) − g(x)k≤2 < 2 <ε/2). Further, let K1 = {x ∈ K| d(x, B) ≥ 1}. By the compactness of K1 and by (∗), con- n clude that f(K1) ⊂ intRn (C ∩ R ) for some n ≥ k0. Using f(K1) ∩ f(B)=∅, note that pk1 (f(B)) ∩ pk1 (f(K1)) = ∅ for some k1 >n. Enlarge B to an open set U ⊂ K to have pk1 (f(U)) ∩ pk1 (f(K1)) = ∅, and let K2 =(K \ U) ∪{x ∈ | ≥ 1 } K d(x, B) 2 . It follows that, for such k1 >k0,

k1 f(K1) ⊂ intRk1 (C∩R ) and pk1 (f(K1))∩pk1 (f(K\int(K2))) = ∅.

Continue this process inductively and observe that conditions (6n) and(7n) hold with g0 replaced by f. 1 \ Let εn = 3 min(d(pkn (f(Kn)),pkn (f(K int(Kn+1))),d(f(Kn+2), kn+2 R \ C)) and ε0 = ε/2. Pick a continuous function ε : K → −1 [0, 1] with ε (0) = B and ε(x) < min(ε0,...,εn+2) for x ∈ ∞ Kn \ Kn−1. Lemma 4 provides a map g0 : K → R , g0|K \ B ∈ a H (K \ B) such that kg0(x) − f(x)k≤ε(x) <ε/2, x ∈ K. It is evident that g0 satisﬁes (4) and (5). For x ∈ Kn \ Kn−3, n nk f(x) ∈ intR k (C ∩ R ); hence pnk (f(x)) = f(x) and

kpnk (g0(x)) − f(x)k = kpkn (g0(x)) − pkn (f(x))k

≤kg0(x) − f(x)k

≤ ε(x) <εn−2.

≤ 1 kn \ ∈ Since εn−2 3 d(f(Kn), R C), we conclude that pkn (g0(x)) kn intRkn (C ∩ R ), showing (6n). To verify (7n), ﬁx y ∈ K \ int(Kn+1) and let x ∈ Km \ Km−1 ⊂ Kn for some 1 ≤ m ≤ n.

Then, we have 3εm ≤kpkm (f(x)) − pkm (f(y))k, and ε(x),ε(y) < εm. Therefore,

kpkn (g0(x)) − pkn (g0(y)k≥kpkm (g0(x)) − pkm (g0(y))k≥

kpkm (f(x)) − pkm (f(y))k−kpkm (f(x)) − pkm (g0(x))k

−kpkm (f(y)) − pkm (g0(y))k≥ 88 Tadeusz Dobrowolski

3εm−kf(x)−g0(x)k−kf(y)−g0(y)k≥3εm−ε(x)−ε(y)≥εm > 0.

By (6n) and (7n), g0(B) ∩ g0(K \ B)=∅; this together with the a fact that, g0|K \B ∈ H (K \B), yields that g0 is an embedding. This ﬁnishes the construction of g0. Suppose that embeddings g0,...,gn−1 have been constructed so that (1n)–(3n), n ≥ 1, are satisﬁed. Suppose additionally that

kn+1 ◦ \ ∈ k ∩ (8n−1) pkn+1 gn−1(Kn Kn−2) intR n+1 (C R )

(9n−1) pkn+1 ◦ gn−1(Kn \ int(Kn−1)) ∩ pkn+1 ◦ gn−1(Kn−2 ∪ (K \ int(Kn+1))) = ∅; are satisﬁed. (Notice that (80) follows from (60), for K1 \ K−1 ⊂ K2 \ K−1, and that (90) is a direct consequence of (70), for if the sets in (90) have a point in common then the sets in (70) do as well). ∞ We are ready to deﬁne a required embedding gn : K → R . a a ∞ Since gn−1|K \ B ∈ H (K \ B), there exists h ∈ H (R ) such that gn−1|Kn = h|Kn; hence, by Lemma 1, for some β ≥ 1,

gn−1(Kn) ⊆ h(Kn) ⊆ βX and gn−1(Kn ∩ C) ⊆ βC.

By (8n−1) there exists 0 <δ<1 with pkn+1 ◦gn−1(Kn \int(Kn−1)) ⊆ − δ (1 2 )C. Employing (9n−1), pick a continuous function kn+1 → δ ∈ ◦ ∪ α : R [ 2β , 1] such that α(x)=1forx pkn+1 gn−1(Kn−2 \ δ ∈ ◦ \ (K int(Kn+1))) and α(x)= 2β for x pkn−1 gn−1(Kn int(Kn−1)), set

kn+1 h(x)=pkn+1 (x)+α(pkn+1 (x))p (x)

= α(pkn+1 (x))x +(1− α(pkn+1 (x)))pkn+1 (x), and let gn = h ◦ gn−1. It is clear that gn is an embedding a such that gn|K \ B ∈ H (K \ B). Evidently, (2n) and (3n) are satisﬁed. To verify the ﬁrst assertion of (1n), let x ∈ Kn; if x ∈ Kn−2 then the assertion follows from (2n) and (1n−1); THE STRONG UNIVERSALITY OF CERTAIN STARLIKE ... 89 if x ∈ Kn\Kn−1 then gn−1(x) ∈ rint(X)by(1n−1), pkn+1 (gn−1(x)) kn+1 δ ∈ k ∩ intR n+1 (C R )by(8n−1), and hence gn(x)=2β gn−1(x)+ − δ ∈ δ − ∈ (1 2β )pkn+1 (gn−1(x)) 2 X +(1 δ)C rint(X). A similar argument works for the proof of the second part of the assertion.

To verify (8n), we ﬁrst note that gn(x) ∈ [pkn+1 (gn−1(x)) ,gn−1(x)] and gn−1(x) ∈ [pkn (gn−2(x)),gn−2(x)] by (3n) and (3n−1), respectively. Hence, we have

pkn+2 (gn(x)) ∈ [pkn+1 (gn−1(x)),pkn+2 (gn−1(x))] =

[pkn+1 (g0(x)),pkn+2 (g0(x))] and

pkn+2 (gn−1(x)) ∈ [pkn (gn−2(x)),pkn+2 (gn−2(x))] =

[pkn (g0(x)),pkn+2 (g0(x))]

(as well as pkn+1 (gn−1(x)) ∈ [pkn (g0(x)),pkn+1 (g0(x))]) for x ∈ K \ Kn and x ∈ K \ Kn−1, respectively, because, in these cases, gn−1(x)=g0(x) and gn−2(x)=g0(x)by(2n−1)–(21).

It follows that pkn+2 (gn(x)) ∈ [pkn+1 (g0(x)),pkn+2 (g0(x))] and pkn+2 (g0(x)) ∈ conv(pkn (g0(x)),pkn+1 (g0(x)),pkn+2 (g0(x))) for x ∈ K \ Kn and x ∈ K \ Kn−1, respectively. Now, pick x ∈ Kn+1 \ Kn−1;ifx ∈ Kn+1 \ Kn ⊂ K \ Kn [resp., if x ∈ Kn \ Kn−1 ⊂ K \ Kn−1] then an application of (6n) and (6n−1) [resp., (6i) for i = n, n +1,n+ 2] yields (8n) (use the fact that Kn \ Kn−1 is contained in each of Kn \ Kn−3, Kn+1 \ Kn−2, and Kn+2 \ Kn−1). To show (9n), notice that for x ∈ Kn+1 \ int(Kn) we have pkn+1 (gn(x)) = pkn+1 (g0(x)) by an application of (3n) and (2n−1)–(21); also, if y ∈ Km \ Km−1 ⊂ Kn,1≤ m ≤ n − 1, we have pkm (gn(y)) = pkm (gm−1(y)) = pkm (g0(y)), and if y ∈ K \ int(Kn+2) then pkn+1 (gn(y)) = pkn+1 (g0(y)). Now, if

(9n) were not true, i.e., if pkn+2 (gn(x)) = pkn+2 (gn(y)) for some x ∈ Kn+1 \ int(Kn) and y ∈ Kn ∪ (K \ int(Kn+2)), then either pkm (g0(y)) = pkm (g0(x)) (if y ∈ Km \ Km−1)orpkn+1 (g0(x)) = pkn+1 (g0(x)) (if y ∈ K \int(Kn+2)), which contradicts either (7m) or (7n+1), respectively. 90 Tadeusz Dobrowolski 2. Strong Universality for Sets with Almost Internal Points Recall that, for a convex set C,0∈ C is almost internal (see [BP, p.160]) if the set A = {a ∈ C|∃(ε>0) [−εa ∈ C]} is dense in C. Lemma 6. Let C be an infinite-dimensional relatively compact convex subset of a Fréchet space having 0 ∈ C as an almost internal point, and let X, X ⊆ C¯, be such that ker(X)=C. For every compact set K ⊆ X with K ∩span(C)=K ∩C, every B¯ = B ⊆ X, every map f : K → X with f(K \ B) ∩ f(B)=∅, f|B is injective, and f −1(C)∩B = B ∩C, and every ε>0 there exists an embedding f¯ : K → X satisfying (i) f¯|B = f|B; (ii) d(f,f¯ ) <ε; (iii) f¯(K \ B) ⊂ rint(X); (iv) f¯−1(C)=K ∩ C. Proof. By our assumption, C¯ is compact, and there exists a countable set A with the property that, for every a ∈ A, −εa ∈ C for some ε>0. In what follows we adopt an argument of [Ba, Lemma 9] that is a variation of a reasoning from [BP, p. ∗ ∞ 160]. There exists a biorthogonal sequence {(xn,xn)}n=1 (i.e., ∗ m ∗ xm(xn)=δn ) such that span{xn| n ≥ 1} = span(A) and {xn} is total when restricted to the Fréchet space E = span(A). It fol- ∗ lows that T (x)=(xm(x)), x ∈ E, is a continuous linear operator into R∞ whose restriction to C¯ yields an affine embedding onto ¯ ∞ T (C). Moreover, T (C) is an almost convex body relative Rf ∞ (this is because T (A), a subset of T (C ∩span(A)) = T (C)∩Rf , is dense in T (C)). Clearly, ker(T (X)) = T (C). Hence, we can identify T (X) and T (C) with X and C, respectively, and apply Lemma 5 to obtain a required approximation f¯. To show (iv) use Lemma 5(iv)-(v) (if f¯(x) ∈ C for some x ∈ K \ B then f¯(x) ∈ span(C), hence x ∈ K ∩ span(C)= K ∩ C). THE STRONG UNIVERSALITY OF CERTAIN STARLIKE ... 91 Lemma 7. Let C be an infinite-dimensional convex subset of aFréchet space having 0 ∈ C as an almost internal point, and let X, X ⊆ C¯ be such that ker(X)=C. For every compact set L ⊆ X there exists a relatively compact infinite-dimensional convex set D ⊂ C such that (i) 0 is an almost internal point of D; (ii) D = D¯ ∩ C is dense in D¯ ∩ X = X0; (iii) L ⊆ X0 ⊆ ker(X0)=D; (iv) K ∩ D = K ∩ C for every compact set K ⊆ L; (v) span(D) ⊆ span(C) and, consequently, for K ⊆ L with K ∩ span(C)=K ∩ C, we have K ∩ span(D)=K ∩ D. ¯ Proof. Take a sequence A = {an}⊂C so that L ⊆ A, for −n every n there exists εn > 0, εn < 2 , such that −εnan ∈ C and A¯ is infinite-dimensional and compact. Then D = conv(A∪ {−εnan| n ≥ 1}) ∩ C satisfies (i)–(v).

Lemma 8. Let C be an inﬁnite-dimensional convex subset of a Fr´echet space having 0 ∈ C as an almost internal point and let X be such that C ⊆ X ⊆ C¯. Every compact set K ⊆ rint(C¯)∩X is a Z-set in X.

Proof. By the fact that the complement of X is locally homotopy negligible in C¯ and by properties of Z-sets, we can assume that C = X = C¯ (we also can assume that C is separable). If C is compact, then using the fact that 0 is an almost internal point of C,by[BP,?], we conclude that K is a Z-set. A similar argument works also in the case where C is locally compact. If C is not locally compact then every compact subset of C is a Z-set.

Recall that a space X is strongly K-universal if for every closed set B ⊆ K and every map f : K → X such that f|B is a Z-embedding can be approximated by Z-embeddings f¯ with f¯|B = f|B. This approximation is uniform if K is compact, 92 Tadeusz Dobrowolski and uniform with respect to every bounded metric on X, in general. This notion has a relative counterpart, the strong (K, K0)- universality of a pair (X, X0). Namely, assuming additionally that f −1(X0) ∩ B = B ∩ K0, we require that f¯−1(X0)=K0. In the same manner, the strong (K, K0,K00)-universality of (X, X0,X00) is defined (when we write (A, A0)or(A, A0,A00)we always mean that A00 ⊆ A0 ⊆ A). Theorem. Let X be a subset of a Fréchet space such that ker(X)=C is infinite-dimensional having 0 as an almost internal point and X ⊆ C¯. Then, for every compactum K ⊆ X, (i) X is K-universal; (ii) if K ∩C = K ∩span(C), then (X, C) is (K, K ∩C)-strongly universal. Proof. Let us justify (ii), the proof of (i) is similar. Pick a map f : K → X that restricts to some compactum B ⊆ K to a Z-embedding with f −1(C) ∩ B = B ∩ C, and let ε>0. Employing the fact that C is infinite-dimensional, we can additionally assume that f(K \ B) ∩ f(B)=∅. Set L = K ∪ f(K). Apply Lemma 7 to find a relatively compact convex infinite-dimensional set D with properties (i)-(v). Let X0 = D¯ ∩ X. For the triple (f, D, X0) obtain the embedding f¯ : K → X0 satisfying (i)-(iv) of Lemma 6. By (iv) of Lemma 7, it easily follows that f¯(K ∩ C) ⊂ C.If for x ∈ K we have f¯(x) ∈ C ∩ X0 ⊆ C ∩ D¯ , then x ∈ D ⊆ C. Since f¯(K \ B) ⊂ rint(X0) ⊂ rint(D¯ ) ∩ X ⊂ rint(C¯) ∩ X,by Lemma 8, we infer that every compact subset of f¯(K \ B)isa Z-set in X. This together with the fact that f¯(B)isaZ-set in X, gives that f¯(K)isaZ-set in X. Remark 2. The argument of Theorem actually shows that when X and C satisfy the above hypothesis then, for every compactum K ⊆ C¯ such that K ∩ C = K ∩ span(C) and K ∩ X = K ∩span(X), the triple (C,X,C¯ ) is strongly (K, K ∩X, K ∩C)- universal. Using the results [BRZ, 1.7.9 and 5.2.6] (see [BC2]), THE STRONG UNIVERSALITY OF CERTAIN STARLIKE ... 93 we obtain the strong (K ∩ X, K ∩ C) universality of (X, C). In the same manner, we can obtain the strong (K, K ∩ X)- universality of (C,X¯ ) and the strong K ∩ X universality of X for a compactum K ⊆ C¯ such that K ∩ X = K ∩ span(X). Below we provide an example showing the strong universality does not hold, in general, for the class of all closed subsets of a convex set. However, we do not know whether the hypothesis concerning the existence of the almost internal point of C is essential in Theorem and Remark 2. Example 1. Let B be the unit open ball of `2, and let S be the unit sphere, the boundary of B. Let A be an incomplete subset of S. Then C = B ∪A is a convex set that is not strongly universal for A (which is a closed subset of C). Simply, the constant map of A with value 0 ∈ B cannot be approximated by closed embeddings. (Letting, e.g., A = Q be the space of rationals, we see that C could even be obtained as a countable union of closed completely metrizable sets.) This contrasts with the results of [BRZ, 5.3.2 and 5.3.5], where A is assumed to be a closed subset of span(A). Notice also that the strong universality of X discussed in Remark 2 is relative to the class of sets that are closed in span(X); our example shows that the hypothesis ‘of being closed’ is essential. The space C does not have the Zσ-property. To obtain an example with this extra property let C0 =(A0 ×{0}) ∪ (B × B0) ⊂ `2 × `2, where A0 ⊂ 0 2 2 2 S×{0}, B = B∩`f , and `f is the linear subspace of ` consisting 0 of finite sequences. It is clear that C is a convex Zσ-space. It is clear that if A is a space that cannot be represented as a countable union of closed completely metrizable sets (e.g., A = Q∞), then the constant map of A with value 0 ∈ C0 cannot be approximated by closed embeddings. As indicated previously, the group of homeomorphisms Ha(R∞) (and its counterpart Ha(E) for an arbitrary Fréchet space E) is useful in constructing homeomorphisms with some additional properties (such like diffeomorphisms). Any set X satisfying the assumptions of Theorem has the so-called 94 Tadeusz Dobrowolski compact homeomorphism property, i.e., every homeomorphism h : K → L between compacta K and L in X can be extended to a selfhomeomorphism h¯ of X. For the reason stated above, it would be interesting to know whether h¯ can be taken so that h¯|X \ K ∈ Ha(X \ K) and h¯−1|X \ L ∈ Ha(X \ L). This is the case if X is a linear space, see [Do].

3. Applications to an Embedding Problem and Certain Sigma-compact Absorbers Certain convex subsets C of Fr´echet spaces are homeomorphic to convex subsets of `2. This is known when C is a linear space that is either σ-compact [BD] or, more generally, is contained in a σ-compact space [BC1]. The convex case, which we follow below, was treated in [Ba].

Proposition 1. Assume that X is an Fσ subset of a linear space L, a subspace of a Fr´echet space E, and assume that X is contained in a σ-compact subset of E.Ifker(X)=C is inﬁnite- dimensional having 0 as an almost internal point and X ⊂ C¯, 0 0 then X is homeomorphic to a subset X of `2 with ker(X ) dense ¯0 in X . If, moreover, C is an Fσ subset of span(C) then the pairs (X, C) and (X0, ker(X0)) are homeomorphic.

Proof. We we will only take on the absolute case. The case when C¯, the closure of C in E, is locally compact is a consequence of the fact that, according to [BD], there exists an aﬃne ¯ map of C into `2. Assume that C¯ is nonlocally compact. As in [BD], we will show that there exists an injective aﬃne (not necessary continuous) transformation T of X (actually, a σ-compact linear subspace that contains X) into `2 such that X and T (X) are so-called C-absorbers for a certain class C. By the Uniqueness Theorem on absorbers, X and T (X)=X0 are then homeomorphic. To check that X, X0 are C-absorbers we must identify the class C and verify the following conditions: THE STRONG UNIVERSALITY OF CERTAIN STARLIKE ... 95 0 (1) X and X are Zσ-spaces;

(2) X and X0 are countable unions of elements of C;

(3) X and X0 have the strong C-universality property.

We may assume that L = span(X). Find a representation ∞ X = Sn=1 Xn such that Xn is closed in L and Xn ⊆ An for ¯ some compactum An ⊆ C; we may assume that both {Xn} and {An} are increasing sequences. Define C to be the class of home- omorphs of elements of {Xn}. Now, for X, condition (2) is a triviality, and (3) is obtained by an application of Remark 2 to ¯ K = Xn. Condition (1) for X follows from the fact that compacta are Z-sets in nonlocally compact C¯, and the complement of X is locally homotopy negligible in C¯. If we defined T to be continuous on each An then obviously X0 = T (X) would satisfy (2). In order to achieve (1) and (3), we must arrange that T is continuous on some compact enlargements of each An. Take a sequence D = {dn}⊂C such that for every n there exists εn > 0 with −εndn ∈ C.For each n, we can find a set Dn ⊆ D such that Añ, the closure ˜ of Dn, is compact and An ⊆ An. If only T is continuous on ˜ 0 each An then T (D) is dense in T (X)=X ; hence, 0 would be an almost internal point, and Remark 2 yields (3). The following construction of the enlargements is that of [Ba]. By the fact that Xn is a Z-set in X, for each n there exists a homotopy αn : conv{d1,...,dn}×[0, 1] → C such that αn(·, 0) = id and im(αn(·,t)) ∩ Xn = ∅ for all t>0. Write Bn =im(αn). ∗ Letting {xn} to be a sequence of continuous linear functionals ¯ ∗ −n that separate the points of L be such that |xn(x)| < 2 for all ˜ ∗ x ∈ An ∪ Bn, we see that T (x)=(xn(x)) ia a linear transformation that continuously maps each Añ ∪ Bn into `2. Since T (X¯n) ¯ 0 is compact and T (Xn)=T (X) ∩ T (Xn), T (Xn) is closed in X . 0 To show that T (Xn)isaZ-set in X for a given ε>0, use the density of T (D)inX0,toε-approximate a map ϕ : Q → X0 0 by a map ϕ : Q → conv{T (d1),...,T(dm)} for some m ≥ n. 96 Tadeusz Dobrowolski −1 0 Let ϕt(q)=T (αm(T (ϕ (q),t)). By the continuity of T on Bm, 0 there exists t0 > 0 such that ϕt0 is ε-close to ϕ and obviously im(ϕt0 ) misses T (Xm) ⊃ T (Xn).

Remark 3. It is not difficult to make adjustments in the above proof to show that, under respective assumptions of Proposition 1, there exist σ-absorbers Y and Y 0 with the following inclusions ¯ 0 0 ¯0 X ⊆ Y ⊂ C and X ⊆ Y ⊆ X ⊂ `2 such that the pairs (Y,X) and (Y 0,X0) are homeomorphic [resp., the triples (Y,X,C) and (Y 0,X0, ker(X0)) are homeomorphic]. Consider the infinite-dimensional compact convex ellipsoid ∞ 2 2 M = {(xi) ∈ `2| P1 i xi ≤ 1}. Any separable metric space Z embeds onto a linearly independent subset of the pseudobound- ∞ 2 2 ary S = {(xi) ∈ M| P1 i xi =1} of M. One always can find such an embedding so that additionally there exists a countable set D ⊂ S \ Z linearly independent and dense in S. Let

P = span(D) ∩ M, which can obviously be expressed as a countable union of sets homeomorphic with ﬁnite-dimensional cubes. We will consider the following counterpart of the space Ω(Z) introduced in [BC2]

ω(Z)={tz +(1− t)p| z ∈ Z, p ∈ P, 0 ≤ t ≤ 1}⊂M.

Proposition 2. For a sigma-compact space Z, the space ω(Z) is an absorber for C, the class of all compacta embeddable in Z × P ; moreover, the complement of ω(Z) is locally homotopy negligible in M.

∞ ∞ Proof. Express Z = Sn=1 Zn and P = Sn=1 Pn as countable unions of compacta Zn and Pn. For every pair of compacta K ⊂ Z and L ⊂ P , write ω(K, L)={tz +(1− t)p| z ∈ K, p ∈ L, 0 ≤ t ≤ 1}. Since the map (z,p,t) → tz+(1−t)p is injective, ω(K, L) is homeomorphic to K × L × [0, 1]; hence, ω(K, L)isa THE STRONG UNIVERSALITY OF CERTAIN STARLIKE ... 97 ∞ compactum embeddable in Z × P .Asω(Z)=Sn=1 ω(Zn,Pn), we have that ω(Z) is countable union of elements of C. Since the complement of P \ S in M \ S is locally homotopy negligible, ω(Z)isaZσ-space. Note also that the complement of ω(Z) is locally homotopy negligible in M. The strong universality of ω(Z) for the class C follows from Theorem because P ⊆ ker(ω(Z)) ⊆ P¯ = M, and 0 is an almost interior point of P .

Corollary 1. Let Z be that of Proposition 2. For every m, the product ω(Z)m is an absorber for the class of compacta embeddable in Zm × P ; moreover, the complement of ω(Z)m is locally homotopy negligible in M m. Hence, ω(Z)m is homeomorphic to ω(Zm).

Proof. The only thing that needs justiﬁcation is the strong universality property of ω(Z)m for the class of all its compacta. This is, however, a consequence of Theorem because P m ⊂ ker(ω(Z)m) ⊂ M m.

For a set A ⊆ Zm, we say that ϕ : A → Y is ﬁberwise −1 injective if ϕ is injective on each set A∩Fi, where Fi = πi (p), m−1 m m−1 p ∈ Z ,1≤ i ≤ m, and πi : Z → Z is the projection given by πi((zi)) = (z1,...,zi−1,zi+1,...,zm). Lemma 9. For every map Φ:ω(Z)m → ω(Z)n there exist an open set U ⊆ Zm and an embedding i :Φ(j(U)) → Zn × Iq, where j : Zm → ω(Z)m stands for canonical embedding. In particular, if there exists a ﬁberwise injective map (resp. an embedding) Φ:ω(Z)m → ω(Z)n, n

Proof. Identify Zm with a subspace of ω(Z)m. By Corollary 1, Φ(Zm) is covered by countable many compact subsets of Zn ×P . By the Baire Theorem, there exists an open subset U of Zm so that Φ(U) ⊂ Zn × Iq for some q. 98 Tadeusz Dobrowolski In what follows, we closely mimic a reasoning of [BC2, The- orem 6.7]. Let C be the Cook’s continuum [Co]. This is a hereditary indecomposable continuum such that every map of a subcontinuum A of C into C is either a constant map or the inclusion map. We let

∞

K = Y Ai, i=1 where Ai are pairwise disjoint subcontinua of C. For the purpose of Corollary 3, we note that K is strongly inﬁnite-dimensional, see [En] for terminology. Proposition 3. The above compactum K has the property that for no open set U ⊆ Km there exists a ﬁberwise injective map ϕ : U → Kn × Iq for n

Proof. We check our assertion when m = 3 and n = 2 only; a similar argument works for arbitrary n

m (ii) writing Cm for the class of compacta embeddable in K × P , it follows that Cm are pairwise distinct for distinct m’s; hence, Cm is not a multiplicative class, however, it has the property that L × [0, 1] ∈Cm for every L ∈C; (iii) there is no group structure or convex structure on any ω(K)m. Proof. To see (i) combine Lemma 9 and Proposition 3. For (ii), assume Km is embeddable in Kn × P for some n

Example 2. Take a null-sequence {Cn} of pairwise disjoint Cantor sets in the Cantor set C such that every nonempty open subset of C contains some Cn. For every n, pick a map fn of Cn onto Sα. Let Rα be the adjoint space obtaining by attaching Sα to C via maps fn. The compactum R = Rα is as required.

Proof. We will show that trInd R ≤ α + 1. Write for q the quotient map. Let Fn be a finite clopen cover of C with di- ameters < 1/n and such that d(F, F 0) > 1/n for F =6 F 0, 0 F, F ∈Fn. Let A and B be closed disjoint subsets of R. There exists k ∈ N such that d(q−1(A),q−1(B)) > 1/k. Let −1 UA = S{F ∈Fn|A ∩ q (F ) =6 ∅}; in the same way define UB. Then UA and UB are clopen subsets of C with UA ∩ UB = ∅, −1 −1 UA ∪ UB = C, UA ⊃ q (A), and UB ⊃ q (B). It follows that D = q(UA) ∩ q(UB) is a partition between A and B. Since d(UA,UB) > 1/k and diam Cn → 0, there exists m ∈ N such that if Cn has a point in common with both UA and UB then m n ≤ m. We conclude that D ⊂ Sn=1 q(Cn); hence, A is contained in a disjoint finite union of copies of the Smirnov cube Sα. Consequently, trInd D ≤ α, and therefore trInd R ≤ α +1 (and obviously trind R ≤ α + 1).

Proposition 4. For the above compactum R = Rα0 with α0 = ωω, no open set of Rm admits an injective map into Rn × Iq for nα(+)n for arbitrary n. This is, however, a consequence of Chatyrko’s result [Ch] stating that trind(Sα × Sa) = trind(Sα(+)α), which yields that trind(Sα0 × ω ω Sα0 )=α0(+)α0 = ω (+)ω ; here, we use the facts that (1) ω α0(+)α0 is a so-called invariant ordinal, that is, α0(+)α0 = ω ·γ for some γ, and (2) trind Sβ = β for invariant ordinals [Lu].

Corollary 3. The statements (i) and (ii) of Corollary 2 hold when K is replaced by the compactum R. In particular, there is no embedding of ω(R)m and ω(R)n are nonhomeomorphic for n =6 m. Moreover, for every n, ω(R)n is countable-dimensional, and ω(K)m and ω(R)n are nonhomeomorphic for any m and n.

Proof. The ﬁrst statement follows in the same way as (i) and (ii) of Corollary 2. The ‘moreover’ part is a consequence of the facts that Rn is countable-dimensional, and K is strongly inﬁnite-dimenisonal, see [En].

We thank R. Pol and E. Pol for pointing out to us the space R and its properties described above.

There exists yet another example of a compactum that may replace K in Corollary 2(i) when n =1andm = 2. Such a compactum Q involving a hereditary infinite-dimensional continuum was discovered by J. Kulesza. It turns out that the spaces ω(K), ω(R), and ω(Q) are pairwise nonhomeomorphic. It is likely that there are uncountably many such examples. 102 Tadeusz Dobrowolski References [Ba] T. Banakh, Towards to topological classification of convex sets in infinite-dimensional Fréchetspaces, preprint [BC1] T. Banakh and R. Cauty, Universalitéforte pour les sous- ensembles totalement bornés. Colloq. Math. 73 1997, 25–33. [BC2] T. Banakh and R. Cauty, Interplay between strongly universal spaces and pairs, Dissertationes Math., 386 2000, 1–38. [BRZ] T. Banakh, T. Radul and M. Zarichnyi, Absorbing Sets in Infinite-Dimensional Manifolds, VNTL Publishers, Lviv 1996. [BD] C. Bessaga and T. Dobrowolski, Affine and homeomorphic embeddings into `2, Proc. Amer. Math. Soc. 125 1997, 259–268. [BP] C. Bessaga and A. Pelczyński, Selected Topics in Infinite- Dimensional Topology, Polish Scientific Publishers, Warszawa 1975. [Ch] V. A. Chatyrko, Ordinal products of topological spaces, Fund. Math 144 1994, 95–117. [Co] H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 1967, 241–249. [Do] T. Dobrowolski, Extending homeomorphisms and applications to metric linear spaces without completeness, Trans. Amer. Math. Soc. 313 1989, 753–784. [DM1] T. Dobrowolski and J. Mogilski, Sigma-compact locally convex metric linear spaces universal for compacta are homeomorphic, Proc. Amer. Math. Soc. 78 1982, 683–658. [DM2] T. Dobrowolski and J. Mogilski, Problems on topological classification of incomplete metric spaces in Open Problems in Topol- ogy, editors J. van Mill and G.M. Reed, North-Holland, Ams- terdam 1990. [En] R. Engelking, Theory of Dimensions Finite and Infinite, Heldr- mann Verlag, Lemgo, Germany 1995. [Lu] L. A. Luxemburg, On compact metric spaces with noncoinciding transfinite dimensions, Pacific J. Math. 93 1981, 339–386. THE STRONG UNIVERSALITY OF CERTAIN STARLIKE ... 103 [vM] J. van Mill, Infinite-Dimensional Topology: Prerequisites and Introduction, North-Holland Publishing Company, Amsterdam 1989. [Re] M. Reńska, On Cantor manifolds for the large transfinite di- mension, Topology Appl. 112 2001, 1–11 [To] H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of `2-manifolds, Fund. Math 101 1978, 93–110. [Za] M. Zarichnyi, Soft maps and their applications in infinite- dimensional topology, preprint.

Department of Mathematics, Pittsburg State University, Pitts- burg, KS 66762 E-mail address: [email protected]