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... The proof of Theorem 2.6 is based on the technique, developed in Section 2, for transferring properties of metrizable ANR's to LC n -subspaces (in this way well known properties of metrizable LC n -spaces can be obtained from the corresponding properties of metrizable ANR's, see for example Proposition 2.2). Section 2 contains also a characterization of metrizable LC n -spaces whose analogue for ANR's was established by Nhu [12]. ...
... We complete this section by a characterization of metrizable LC n -spaces similar to the characterization of metrizable ANR-spaces provided in [12] (see also [17, Theorem 6.8.1]). Proof. ...
... To prove the other implication, embed M as a closed subset of a metrizable space Z with dim Z\M ≤ n + and follow the proof of implication (iii) ⇒ (i) from [12, Theorem 1.1] to obtain that M is a retract of a neighborhood W of M in Z (the only di erence is that in Fact 1.2 from [12] we take the cover V of W \M to be of order ≤ n + , so the nerve N(V) is a complex of dimension ≤ n + ). Then by [13, chapter V, Theorem 3.1], M is LC n . ...
We provide a machinery for transferring some properties of metrizable ANR-spaces to metrizable
LCn-spaces. As a result, we show that for completely metrizable spaces the properties ALCn, LCn and
WLCn coincide to each other. We also provide the following spectral characterizations of ALCn and celllike
compacta: A compactum X is ALCn if and only if X is the limit space of a σ-complete inverse system S = {Xα , pβ
α , α < β < τ} consisting of compact metrizable LCn-spaces Xα such that all bonding projections
pβα, as a well all limit projections pα, are UVn-maps.
A compactum X is a cell-like (resp., UVn) space if and only if X is the limit space of a σ-complete inverse
system consisting of cell-like (resp., UVn) metrizable compacta.
... The proof of Theorem 2.7 is based on the technique, developed in Section 2, for transferring properties of metrizable ANR's to LC n -subspaces (in this way well known properties of metrizable LC n -spaces can be obtained from the corresponding properties of metrizable ANR's, see for example Proposition 2.2). Section 2 contains also a characterization of metrizable LC n -spaces whose analogue for ANR's was established by Nhu [12]. ...
... We complete this section by a characterization of metrizable LC nspaces similar to the characterization of metrizable ANR-spaces provided in [12] (see also [17, Theorem 6.8.1]). Proof. ...
... To prove the other implication, embed M as a closed subset of a metrizable space Z with dim Z\M ≤ n + 1 and follow the proof of implication (iii) ⇒ (i) from [12, Theorem 1.1] to obtain that M is a retract of a neighborhood W 1 of M in Z (the only difference is that in Fact 1.2 from [12] we take the cover V of W 1 \M to be of order ≤ n + 1, so the nerve N(V) is a complex of dimension ≤ n + 1). Then by [13, chapter V, Theorem 3.1], M is LC n . ...
We provide a machinery for transferring some properties of metrizable
ANR-spaces to metrizable -spaces. As a result, we show that for
complete metrizable spaces the properties , and coincide
to each other. We also provide the following spectral characterizations of
and cell-like compacta: A compactum X is if and only if X
is the limit space of a -complete inverse system consisting of compact metrizable
-spaces such that all bonding projections ,
as a well all limit projections , are -maps. A compactum X is
a cell-like (resp., ) space if and only if X is the limit space of a
-complete inverse system consisting of cell-like (resp., ) metric
compacta.
... Such subfunctors were considered in [165] and [324] for F = exp. For any compactum X the set F#u also has a stronger topology, the topology of right limits Iim{FnX}. ...
... We should note that the existence of the natural mapping p : 3) FXeAR.c~-F*XoAR-*~-X 9 is a Peano continuum (Eberhart,Nadler,and Nowell [191] (Ganea [202] for finite-dimensional X, Jaworowski [242] for all X), SPa n Qaworowski It was proved in [324] that the functors expn and exp,, preserve the class of ANR (~2) spaces. 2. A. N. Dranishnikov is responsible for an elegant proof of a theorem on preservation of the class of AE(n)-spaces. ...
... V. I. Golov [16] showed that under these conditions rx~-tz. A. G. Savchenko proved analogous results (including a fiberwise variant) for the functors expn c [83]. It was shown in [324] that SP~nlg~lz, from which it follows immediately that the functors sP~n preserve/z-manifolds. A rather general result for functors with finite support is given in [26]. ...
This survey is devoted to the properties of certain concrete covariant functors-normal and almost normal functors-in the category of compacta, as well as the algebraic theory of covariant functors, and the connections between the theory of functors with absolute extensors and manifolds.
... To do this, we will first modify slightly the rigid space constructed in [5], and then employ a characterization of ANR-spaces (absolute neighborhood retract spaces) due to the first author, which appeared in its original form in [6] and in a refined form in [7]. It is this second version that we will apply in this paper. ...
... We use the following version of the ANR-characterization theorem; see [6], [7], [8]. ...
A rigid space is a topological vector space whose endomorphisms are all simply scalar multiples of the identity map. This is in sharp contrast to the behavior of operators on l(2), and so rigid spaces are, from the viewpoint of functional analysis, fundamentally different from Hilbert space. Nevertheless, we show in this paper that a rigid space can be constructed which is topologically homeomorphic to Hilbert space. We do this by demonstrating that the first complete rigid space can be modified slightly to be an AR-space (absolute retract), and thus by a theorem of Dobrowolski and Torunczyk is homeomorphic to l(2).
... Our proof also uses the following characterization of ANR-spaces established in [6]: ...
... The following characterization of ANR-spaces was established in [6], see also [8]. ...
Roberts constructed a linear metric space which contains a compact convex set without any extreme points. The space constructed by Roberts is complicated and special.We investigate the topological property of Roberts' example and demonstrate that the linear metric space constructed by Roberts is an AR, therefore is homeomorphic to Hilbert space.
... Hence the above condition implies the condition (iii) of [N,. Thus the theorem follows from [N, Theorem 1-1]. ...
It is shown that any sigma-compact metrizable space is an AR (ANR) if and only if it is (locally) equi-connected and has the compact (neighborhood) extension property.
... The set F n (W ) is an open subset of F n (X) and, since X is an ANR, F n (X) is an ANR for all n ∈ N ( [19]). ...
Let U be an open subset of a locally compact metric ANR X and let be a continuous map. In this paper we study the fixed point index of the map that f induces in the n-symmetric product of X, . This index can detect the existence of periodic orbits of of f, and it can be used to obtain the Euler characteristic of the n-symmetric product of a manifold X, . We compute for all orientable compact surfaces without boundary.
... In our case we use a different approach which involves a characterization of ANR-spaces established by the first author in [N\]. Our proof uses an idea of [N\], see also [Nl,N2], however in our case masses of probability measures lead to more complicated situations. ...
Let Pk(X) denote the set of all probability measures on a metric space X whose supports consist of no more than k points, equipped with the Fedorchuk topology. We prove that if X ∈ ANR then Pk(X) ∈ ANR for every k ∈ N. This implies that for each k ∈ N the functor pk preserves the topology of separable Hubert space.
... Probably the most well-known is Dugundji–Lefschetz' theorem about realizations of polytopes. Another result in this spirit is due to Nhu [7]. We introduce a metric property (Property (B) below) which, roughly speaking, says that there is a sequence of maps of CW-polytopes with some 'compatibility' conditions, related to the metric. ...
We characterize complete metric absolute (neighborhood) retracts in terms of existence of certain maps of CW-polytopes. Using our result, we prove that a compact metric space with a convex and locally convex simplicial structure is an AR. This answers a question of Kulpa from [Topology Proc. 22 (1997) 211–235]. As another application, we prove that the hyperspace of closed subsets of a separable Banach space endowed with the Wijsman topology is an absolute retract.
... Thereby Jaworowski proved Theorem 1.1 for any finite subgroup G < S n and any compact metric G-ANE space X n . Jaworowski's paper is concerned with a special topic and contains a gap [7,8] (which can easily be fixed, though); however, the potential of the approach is very high. Thus, from the standpoint of the modern theory of equivariant extensors, there is no difficulty in making Jaworowski's argument rigorous enough and obtaining a very elegant proof of Theorem 1.1. ...
The group of measure preserving transformations of the unit interval equipped with the weak topology is an absolute retract, hence is homeomorphic to a separable Hubert space.
Every needle point contains a compact convex AR-set without any extreme points. In particular the following spaces contain such compact convex sets: (i) the spaces Lp, 0 ≤ p < 1; (ii) the linear metric space constructed by Roberts (Studia Math. 60 (1977), 255-266).
A subset A of a space X is called a retract of X if there is a map r : X→A such that r | A = id, which is called a retraction. As is easily observed, every retract of a space X is closed in X. A neighborhood retract of X is a closed set in X that is a retract of some neighborhood in X. A metrizable space X is called an absolute neighborhood retract (ANR) (resp. an absolute retract (AR)) if X is a neighborhood retract (or a retract) of an arbitrary metrizable space that contains X as a closed subspace. A space Y is called an absolute neighborhood extensor for metrizable spaces (ANE) if each map f : A → Y from any closed set A in an arbitrary metrizable space X extends over some neighborhood U of A in X. When f can always be extended over X (i.e., U = X in the above), we call Y an absolute extensor for metrizable spaces (AE). As is easily observed, every metrizable ANE (resp. a metrizable AE) is an ANR (resp. an AR). As will be shown, the converse is also true. Thus, a metrizable space is an ANE (resp. an AE) if and only if it is an ANR (resp. an AR).
In this second part of our paper, \ve apply the result of Part 1 to show that the compact convex set with no extreme points, constructed by Roberts (1977), is an AR.
We prove that the original compact convex set with no extreme points, constructed by Roberts (1977) is an absolute retract, therefore is homeomorphic to the Hubert cube. Our proof consists of two parts. In this first part, we give a sufficient condition for a Roberts space to be an AR. In the second part of the paper, we shall apply this to show that the example of Roberts is an AR.
Multivalued fractals are considered as fixed-points of certain induced union operators, called the Hutchinson–Barnsley operators, in hyperspaces of compact subsets of the original spaces endowed with the Hausdorff metric. Various approaches are presented for obtaining the existence results, jointly with the information concerning the topological structure of the set of multivalued fractals. According to the applied fixed-point principles, we distinguish among metric, topological and Tarski’s multivalued fractals. Finite families of condensing and (locally) compact maps as well as of different sorts of contractions are examined with this respect. In particular, continuation principle for multivalued fractals is established for (locally) compact maps. Multivalued fractals are also generated implicitly by means of differential inclusions. A randomization of the deterministic results is indicated. Numerical aspects of computer generated multivalued fractals are discussed in detail.
In this paper, considering the problem when the completion of a metric ANR X is an ANR and X is homotopy dense in the completion, we give some sufficient conditions. It is also shown that each uniform ANR is homotopy dense in any metric space containing X isometrically as a dense subset, and that a metric space X is a uniform ANR if and only if the metric completion of X is a uniform ANR with X a homotopy dense subset. Furthermore, introducing the notions of densely (local) hyper-connectedness and uniformly (local) hyper-connectedness, we characterize of AR's (ANR's) and uniform AR's (uniform ANR's), respectively.
Let X be a locally compact metric absolute neighbourhood retract for metric spaces, U subset of X be an open subset and f:U -->X be a continuous map. The aim of the paper is to study the fixed point index of the map that f induces in the hyperspace of X. For any compact isolated invariant set, K subset of U, this fixed point index produces, in a very natural way, a Conley-type (integer valued) index for K. This index is computed and it is shown that it only depends on what is called the attracting part of K. The index is used to obtain a characterization of isolating neighbourhoods of compact invariant sets with non-empty attracting part. This index also provides a characterization of compact isolated minimal sets that are attractors.
CONTENTSIntroduction § 1. Characterization of n-soft maps of bicompacta § 2. General questions in the theory of covariant functors § 3. Zero-soft and one-soft maps of bicompacta § 4. Non-locally compact absolute extensors § 5. On the properties of maps that are nearly n-soft § 6. The geometry of covariant functorsReferences
We write 2x for the hyperspace of all non-empty compact sets in a complete metric linear space X topologized by the Hausdorff metric. Using the notation F(X) = {A ε{lunate} 2X: A is finite}, lf2 = {x} = (xi) ε{lunate} l2: xi = 0 for almost all i}, and lσ2 = {x = (x i) ε{lunate} l2:σ∞i=1 (ixi)2 < ∞}, we have the following theorem:. A family G ⊂F(X) is homeomorphic to lf2 if G is σ-fd-compact, the closure G of G in 2x is not locally compact and if whenever A, B ∈ G, λ ∈ [0, 1] and C ⊂ λA + (1 - λ)B with card C≤ max{card A, card B} then C ε{lunate} G. Moreover, for any Gδ-AR-set GG of G with GG ⊃ G we have (GG, G)≅(l2, lf{hook}2). Similar conditions for hyperspaces to be homeomorphic to lσ2 are also established.
We introduce the notion of the finite dimensional approximation property (the FDAP) and prove that if a subset X of a linear metric space has the FDAP, then every non-empty convex subset of X is an AR.
As an application we show that every needle point space X contains a dense linear subspace E with the following properties:
(i) E contains a non-empty compact convex set with no extreme points;
(ii) all non-empty convex subsets of E are AR.
By , we denote the space of all closed sets in a space X (including the empty set ∅) with the Fell topology. The subspaces of consisting of all compact sets and of all finite sets are denoted by and , respectively. Let Q=[−1,1]ω be the Hilbert cube, B(Q)=Q⧹(−1,1)ω (the pseudo-boundary of Q) and In this paper, we prove that is homeomorphic to (≈) Q if and only if X is a locally compact, locally connected separable metrizable space with no compact components. Moreover, this is equivalent to . In case X is strongly countable-dimensional, this is also equivalent to .
La presente Memoria tiene por objeto la construcción y el estudio de cierto tipo de índices asociados a los conjuntos compactos, invariantes y aislados de sistemas dinámicos discretos. Estos índices, de propiedades análogos a las del índice de Conley, nos permitirán obtener información sobre la dinámica en los conjuntos mencionados. Dividimos el contenido de este texto en tres partes. En la primera de ellas (Capítulo I) construimos el índice "shape" asociado a un compacto invariante y aislado de un sistema dinámico discreto definido sobre un espacio métrico. Lo novedoso de esta construcción es que prescinde de la condición de compacidad local del espacio, resolviendo así un problema planteado por M.Mrozek. En la segunda parte (Capítulos II y III) asociamos a un compacto invariante y aislado de un sistema dinámico discreto f en un ANR localmente compacto X, los índices de punto fijo de las aplicaciones inducidas por f en ciertos espacios de X. Calcularemos sus valores y veremos cuál es su significado dinámico. Sea f: U C R2 -: R2 un homeomorfismo sobre la imagen y sea K un compacto invariante, aislado y conexo. La tercera y última parte de este estudio (Capítulo IV) calcula el índice de punto fijo, en los entornos aislantes de K, de las interaciones de f, iR2(fk,U(k)). Presentamos un teorema, demostrado de manera simple y geométrica, que generaliza un resultado de P. Le Calvez y J.C. Yoccoz. Obtenemos como corolario la no existencia de homeomorfismos minimales en ciertos subconjuntos de S2.
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