scholarly journals There is no compact metrizable space containing all continua as unique components

2021 ◽  
pp. 107742
Author(s):  
Benjamin Vejnar
Keyword(s):  
Metrizable Space ◽  
Compact Metrizable Space

scholarly journals A function space from a compact metrizable space to a dendrite with the hypo-graph topology

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Hanbiao Yang ◽  
Katsuro Sakai ◽  
Katsuhisa Koshino
Keyword(s):  
Finite Number ◽  
Function Space ◽  
Metrizable Space ◽  
Vietoris Topology ◽  
Hilbert Cube ◽  
Graph Topology ◽  
Closed Sets ◽  
Continuous Map ◽  
Compact Metrizable Space ◽  
Isolated Points

Abstract Let X be an infinite compact metrizable space having only a finite number of isolated points and Y be a non-degenerate dendrite with a distinguished end point v. For each continuous map ƒ : X → Y , we define the hypo-graph ↓vƒ = ∪ x∈X {x} × [v, ƒ (x)], where [v, ƒ (x)] is the unique arc from v to ƒ (x) in Y . Then we can regard ↓v C(X, Y ) = {↓vƒ | ƒ : X → Y is continuous} as the subspace of the hyperspace Cld(X × Y ) of nonempty closed sets in X × Y endowed with the Vietoris topology. Let be the closure of ↓v C(X, Y ) in Cld(X ×Y ). In this paper, we shall prove that the pair , ↓v C(X, Y )) is homeomorphic to (Q, c0), where Q = Iℕ is the Hilbert cube and c0 = {(xi )i∈ℕ ∈ Q | limi→∞xi = 0}.

Non-Metrizable Uniformities and Proximities on Metrizable Spaces

1973 ◽  
Vol 25 (5) ◽  
pp. 979-981
Author(s):  
P. L. Sharma
Keyword(s):  
Metrizable Space ◽  
Compact Metrizable Space ◽  
Metrizable Spaces

In the literature there exist examples of metrizable spaces admitting nonmetrizable uniformities (e.g., see [3, Problem C, p. 204]). In this paper, this phenomenon is presented more coherently by showing that every non-compact metrizable space admits at least one non-metrizable proximity and uncountably many non-metrizable uniformities. It is also proved that the finest compatible uniformity (proximity) on a non-compact non-semidiscrete space is always non-metrizable.

ALMOST MULTIPLICATIVE MORPHISMS AND THE CUNTZ ALGEBRA ${\mathcal O}_2$

1995 ◽  
Vol 06 (04) ◽  
pp. 625-643 ◽  
Author(s):  
HUAXIN LIN ◽  
N. CHRISTOPHER PHILLIPS
Keyword(s):  
Tensor Product ◽  
Metrizable Space ◽  
Cuntz Algebra ◽  
Direct Sums ◽  
Linear Map ◽  
Compact Metrizable Space ◽  
Metrizable Spaces ◽  
Unital Simple

Let X be a compact metrizable space, and let A be a purely infinite simple C*-algbera A satisfying K0(A)=K1(A)=0. We show that an almost multiplicative contractive unital *-preserving linear map from C(X) can be approximated by a homomorphism. As a consequence, we show that if a unital simple C*-algbera [Formula: see text], with [Formula: see text] (finite direct sums), for compact metrizable spaces Xm,n and are even algebras [Formula: see text], satisfies K0(A)=K1(A)=0, then [Formula: see text]. In particular, we show that the tensor product of a simple unital AH-algebra with [Formula: see text] is isomorphic to [Formula: see text].

scholarly journals Isometries of Spaces of Radon Measures

2017 ◽  
Vol 2017 ◽  
pp. 1-4
Author(s):  
Marek Wójtowicz
Keyword(s):  
Closed Subset ◽  
Unit Interval ◽  
Metrizable Space ◽  
Discrete Space ◽  
Hausdorff Spaces ◽  
Bernstein Type ◽  
Radon Measures ◽  
Compact Metrizable Space

Let Ω and I denote a compact metrizable space with card(Ω)≥2 and the unit interval, respectively. We prove Milutin and Cantor-Bernstein type theorems for the spaces M(Ω) of Radon measures on compact Hausdorff spaces Ω. In particular, we obtain the following results: (1) for every infinite closed subset K of βN the spaces M(K), M(βN), and M(Ω2ℵ0) are order-isometric; (2) for every discrete space Γ with m≔card(Γ)>ℵ0 the spaces M(βΓ) and M(I2m) are order-isometric, whereas there is no linear homeomorphic injection from C(βT) into C(I2m).

scholarly journals Which groups act distally?

1983 ◽  
Vol 3 (2) ◽  
pp. 167-185 ◽  
Author(s):  
Herbert Abels
Keyword(s):  
Metrizable Space ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Metrizable Space ◽  
Locally Compact Topological Groups

AbstractThe following question is discussed: which locally compact topological groups have an effective distal action on some compact metrizable space?

scholarly journals Fiberwise KK-equivalence of continuous fields of C*-algebras

2008 ◽  
Vol 3 (2) ◽  
pp. 205-219 ◽  
Author(s):  
Marius Dadarlat
Keyword(s):  
Compact Space ◽  
Metrizable Space ◽  
Cuntz Algebra ◽  
Hilbert Cube ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Finite Dimensional ◽  
Compact Metrizable Space ◽  
Continuous Fields

AbstractLet A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element σ of the parametrized Kasparov group KKX(A,B) is invertible if and only all its fiberwise components σx ∈ KK(A(x),B(x)) are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra . Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.

scholarly journals Pseudo-orbit tracing and algebraic actions of countable amenable groups

2018 ◽  
Vol 39 (9) ◽  
pp. 2570-2591
Author(s):  
TOM MEYEROVITCH
Keyword(s):  
Finite Groups ◽  
Amenable Group ◽  
Countable Group ◽  
Metrizable Space ◽  
Algebraic Characterization ◽  
Positive Entropy ◽  
Algebraic Action ◽  
Compact Metrizable Space ◽  
Algebraic Actions

Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved that they do. We provide examples showing that Chung and Li’s result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li’s algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.

scholarly journals Topological n-cells and Hilbert cubes in inverse limits

2018 ◽  
Vol 19 (1) ◽  
pp. 9
Author(s):  
Leonard R. Rubin
Keyword(s):  
Set Theory ◽  
Weak Topology ◽  
Inverse Limit ◽  
Metrizable Space ◽  
Inverse System ◽  
Inverse Limits ◽  
Hilbert Cube ◽  
Compact Metrizable Space ◽  
Metric Topology

<p>It has been shown by S. Mardešić that if a compact metrizable space X has dim X ≥ 1 and X is the inverse limit of an inverse sequence of compact triangulated polyhedra with simplicial bonding maps, then X must contain an arc.  We are going  to prove that  if X = (|K<sub>a</sub>|,p<sup>b</sup><sub>a</sub>,(A,)<a href="http://www.codecogs.com/eqnedit.php?latex=\preceq" target="_blank"><img title="\preceq" src="http://latex.codecogs.com/gif.latex?\preceq" alt="" /></a>)is an inverse system in set theory of triangulated polyhedra|K<sub>a</sub>|with simplicial  bonding  functions p<sup>b</sup><sub>a</sub> and X = lim X,  then  there  exists  a uniquely determined sub-inverse system X<sub>X</sub>= (|L<sub>a</sub>|, p<sup>b</sup><sub>a</sub>|L<sub>b</sub>|,(A,<a href="http://www.codecogs.com/eqnedit.php?latex=\preceq" target="_blank"><img title="\preceq" src="http://latex.codecogs.com/gif.latex?\preceq" alt="" /></a>)) of X where for each a, L<sub>a</sub> is a subcomplex of K<sub>a</sub>, each p<sup>b</sup><sub>a</sub>|L<sub>b</sub>|:|L<sub>b</sub>| → |L<sub>a</sub>| is  surjective,  and lim X<sub>X</sub> = X. We shall use this to generalize the Mardešić result by characterizing when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain a topological n-cell and do the same in the case of an inverse system of finite triangulated polyhedra with simplicial bonding maps. We shall also characterize when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain an embedded copy of the Hilbert cube. In each of the above settings, all the polyhedra have the weak topology or all have the metric topology(these topologies being identical when the polyhedra are finite).</p>

Embedding -Like Compacta in Manifolds

1967 ◽  
Vol 19 ◽  
pp. 321-332 ◽  
Author(s):  
Michael C. McCord
Keyword(s):  
Inverse Image ◽  
Metrizable Space ◽  
Open Cover ◽  
Compact Metrizable Space

A compactum is a compact, metrizable space. A continuum is a connected compactum. All polyhedra will be finitely triangulable spaces. If a is an open cover of a compactum X, a map of X onto a compactum Y is called an α-map provided that the inverse image of each point in Y is contained in some member of α.If is a class of polyhedra, then, following Mardešić and Segal (10), we say a compactum X is -like provided that for each open cover α of X there exists an a-map of X onto some member of .

On the Representation of Mappings of Compact Metrizable Spaces as Restrictions of Linear Transformations

1970 ◽  
Vol 22 (2) ◽  
pp. 372-375 ◽  
Author(s):  
Michael Edelstein
Keyword(s):  
Hilbert Space ◽  
Linear Transformation ◽  
Continuous Mapping ◽  
Metrizable Space ◽  
Linear Transformations ◽  
Compact Metrizable Space ◽  
One To One ◽  
Metrizable Spaces

Let f: X → X be a continuous mapping of the compact metrizable space X into itself with a singleton. In [3] Janos proved that for any λ, 0 < λ < 1, a metric ρ compatible with the topology of X exists such that ρ(f(x), f(y)) ≦ λρ(x, y) for all x, y ∈X. More recently, Janos [4] has shown that if, in addition, f is one-to-one, then a Hilbert space H and a homeomorphism μ: X → H exist such that μfμ-1 is the restriction to μ[X] of the transformation sending y ∈ H into λy. Our aim in this note is to show that in both cases a homeomorphism h of X into l2 exists such that hfh-1 is the restriction of a linear transformation.

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