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}.

A Hilbert cube compactification of the function space with the compact-open topology

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Atsushi Kogasaka ◽  
Katsuro Sakai
Keyword(s):  
Closed Interval ◽  
Continuous Functions ◽  
Metrizable Space ◽  
Hilbert Cube ◽  
Compact Open Topology ◽  
Locally Compact ◽  
Closed Sets ◽  
Isolated Points ◽  
Compact Case ◽  
Separable Metrizable Space

AbstractLet X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $$ \bar C $$(X) of C(X) such that the pair ($$ \bar C $$(X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification $$ \bar C $$(X) coincides with the space USCCF(X,) of all upper semi-continuous set-valued functions φ: X → = [−∞, ∞] such that each φ(x) is a closed interval, where the topology for USCCF(X, ) is inherited from the Fell hyperspace Cld*F(X × ) of all closed sets in X × .

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 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>

scholarly journals The Metrizability of Spaces whose Diagonals have a Countable Base

1977 ◽  
Vol 20 (4) ◽  
pp. 513-514 ◽  
Author(s):  
John Ginsburg
Keyword(s):  
Metrizable Space ◽  
Countable Base ◽  
Neighborhood Base ◽  
Isolated Points

AbstractIt is shown that the diagonal of X has a countable neighborhood base in X × X if and only if X is a metrizable space whose set of non-isolated points is compact.

A Remark on Invariant Scrambled Sets

2012 ◽  
Vol 204-208 ◽  
pp. 4776-4779
Author(s):  
Lin Huang ◽  
Huo Yun Wang ◽  
Hong Ying Wu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Periodic Point ◽  
Metric Space ◽  
Cantor Sets ◽  
Countable Union ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Isolated Points ◽  
Weakly Mixing

By a dynamical system we mean a compact metric space together with a continuous map . This article is devoted to study of invariant scrambled sets. A dynamical system is a periodically adsorbing system if there exists a fixed point and a periodic point such that and are dense in . We show that every topological weakly mixing and periodically adsorbing system contains an invariant and dense Mycielski scrambled set for some , where has no isolated points. A subset is a Myceilski set if it is a countable union of Cantor sets.

scholarly journals SELF ORGANIZING NEURAL MAPS IN THE PROBLEMS OF ECOLOGICAL MONITORING

2021 ◽  
pp. 112-117
Author(s):  
O. Getmanets ◽  
M. Pelikhatyi
Keyword(s):  
Finite Number ◽  
Environmental Pollution ◽  
Background Radiation ◽  
Ecological Monitoring ◽  
Abiotic Factor ◽  
Radiation Dose Rate ◽  
Continuous Map ◽  
And Cluster Analysis ◽  
Neural Maps ◽  
Self Organizing

There is a certain problem in ecological monitoring of the environment state according to the measured values of a certain abiotic factor. Namely, how to build a continuous map of environmental pollution throughout the controlled area, based on the results of measurements carried out at a finite number of points inside the controlled territory. The aim of the work is to study the possibility of using the method of self organizing neural maps (SOM) for the problems of the ecological monitoring of the environment, and specifically for building an accurate continuous map of environmental pollution on the ground. The materials and methods of researches are the results of measurements the ambient equivalent of the continuous X-ray and gamma radiation dose rate on a territory of the historical center of Kharkiv has been used as research materials; processing of the obtained data by SOM's methods using MatLab 8.1 and STATISTICA 10 computer programs has been done. Results: in the process of 1000 self-learning cycles of a neural network of 100 initial active neurons randomly located on the controlled area map, 25 neural clusters have been obtained, the coordinates of the centers of which practically coincided with the 25 control points coordinates. A continuous map of the background radiation on the controlled area has been built. The accuracy of this map was no worse than 0.25 μR/hour. Conclusions: the possibility of using the SOM methods to build a continuous map of the level of environmental pollution on the ground based on the results of measuring the values of a certain abiotic factor in a finite number of points has been proven. It has been proven that this method is more accurate compared to the methods of regression mapping and cluster analysis, from which it is essentially different. The possibilities for a significant improvement in the accuracy of the method lie in increasing the number of initial neurons on the terrain map and the number of iterations during their training.

The Number of Closed Subsets of a Topological Space

1978 ◽  
Vol 30 (02) ◽  
pp. 301-314 ◽  
Author(s):  
R. E. Hodel
Keyword(s):  
Topological Space ◽  
Basic Problem ◽  
Cardinal Number ◽  
Metrizable Space ◽  
Infinite Cardinal ◽  
Closed Sets ◽  
Complete Answer ◽  
Metrizable Spaces

Let X be an infinite topological space, let 𝔫 be an infinite cardinal number with 𝔫 ≦ |X|. The basic problem in this paper is to find the number of closed sets in X of cardinality 𝔫. A complete answer to this question for the class of metrizable spaces has been given by A. H. Stone in [31], where he proves the following result. Let X be an infinite metrizable space of weight 𝔪, let 𝔫 ≦ |X|.

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.

Function Space Topologies for Connectivity and Semi-Connectivity Functions

1966 ◽  
Vol 9 (3) ◽  
pp. 349-352 ◽  
Author(s):  
Somashekhar Amrith Naimpally
Keyword(s):  
Uniform Convergence ◽  
Function Space ◽  
Uniform Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Subsequent Paper ◽  
Graph Topology ◽  
Function Space Topologies

Let X and Y be topological spaces. If Y is a uniform space then one of the most useful function space topologies for the class of continuous functions on X to Y (denoted by C) is the topology of uniform convergence. The reason for this usefulness is the fact that in this topology C is closed in YX (see Theorem 9, page 227 in [2]) and consequently, if Y is complete then C is complete. In this paper I shall show that a similar result is true for the function space of connectivity functions in the topology of uniform convergence and for the function space of semi-connectivity functions in the graph topology when X×Y is completely normal. In a subsequent paper the problem of connected functions will be discussed.

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].

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