Cliquishness and Quasicontinuity of Two-Variable Maps

2013 ◽  
Vol 56 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. Bouziad
Keyword(s):  
Wide Class ◽  
Metric Space ◽  
Winning Strategy ◽  
Positive Answer ◽  
Baire Space ◽  
Compact Spaces ◽  
Open Set ◽  
Continuous Maps ◽  
Dense Set

AbstractWe study the existence of continuity points for mappingswhose x-sectionsare fragmentable and y-sectionsare quasicontinuous, where X is a Baire space and Z is a metric space. For the factor Y, we consider two infinite “pointpicking” games G1(y) and G2(y) defined respectively for each y ∈ Y as follows: in the n-th inning, Player I gives a dense set Dn⊂ Y, respectively, a dense open set Dn⊂ Y. Then Player II picks a point yn∈ Dn; II wins if y is in the closure of {yn: n ∈ N}, otherwise I wins. It is shown that (i) f is cliquish if II has a winning strategy in G1(y) for every y ∈ Y, and (ii) f is quasicontinuous if the x-sections of f are continuous and the set of y ∈ Y such that II has a winning strategy in G2(y) is dense in Y. Item (i) extends substantially a result of Debs and item (ii) indicates that the problem of Talagrand on separately continuous maps has a positive answer for a wide class of “small” compact spaces.

ON A NEW CLASS OF SEMI GENERALIZED CLOSED SETS IN STRONG GENERALIZED TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (11) ◽  
pp. 9353-9360
Author(s):  
G. Selvi ◽  
I. Rajasekaran
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Open Set ◽  
Continuous Maps

This paper deals with the concepts of semi generalized closed sets in strong generalized topological spaces such as $sg^{\star \star}_\mu$-closed set, $sg^{\star \star}_\mu$-open set, $g^{\star \star}_\mu$-closed set, $g^{\star \star}_\mu$-open set and studied some of its basic properties included with $sg^{\star \star}_\mu$-continuous maps, $sg^{\star \star}_\mu$-irresolute maps and $T_\frac{1}{2}$-space in strong generalized topological spaces.

On the Dimension Modulo Classes of Topological Spaces and Free Topological Groups

2003 ◽  
Vol 10 (2) ◽  
pp. 209-222
Author(s):  
I. Bakhia
Keyword(s):  
Wide Class ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Topological Groups ◽  
Compact Spaces ◽  
Topological Semigroups

Abstract Functions of dimension modulo a (rather wide) class of spaces are considered and the conditions are found, under which the dimension of the product of spaces modulo these classes is equal to zero. Based on these results, the sufficient conditions are established, under which spaces of free topological semigroups (in the sense of Marxen) and spaces of free topological groups (in the sense of Markov and Graev) are zero-dimensional modulo classes of compact spaces.

scholarly journals On equicontinuous factors of flows on locally path-connected compact spaces

2020 ◽  
pp. 1-18
Author(s):  
NIKOLAI EDEKO
Keyword(s):  
Lie Group ◽  
Metric Space ◽  
Betti Number ◽  
Alternative Proof ◽  
Simple Consequence ◽  
Compact Metric Space ◽  
Invariant Functions ◽  
Compact Spaces ◽  
Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

scholarly journals New results on topological dynamics of antitriangular maps

2001 ◽  
Vol 2 (1) ◽  
pp. 51 ◽  
Author(s):  
Francisco Balibrea ◽  
J.S. Cánovas ◽  
A. Linero
Keyword(s):  
Metric Space ◽  
Topological Dynamics ◽  
Compact Metric Space ◽  
Special Analysis ◽  
Continuous Maps

<p>We present some results concerning the topological dynamics of antitriangular maps, F:X<sup>2</sup>→ X<sup>2 </sup>with the formvF(x,y)=(g(y),f(x)), where (X,d) is a compact metric space and f,g : X→ X are continuous maps. We make an special analysis in the case of X = [0,1].</p>

scholarly journals On Dentability in Locally Convex Vector Spaces

2017 ◽  
Vol 10 ◽  
pp. 14-19
Author(s):  
Oleg Reinov ◽  
Asfand Fahad
Keyword(s):  
Vector Space ◽  
Wide Class ◽  
Positive Answer ◽  
Vector Spaces ◽  
Bounded Subset ◽  
Locally Convex

The notions of V-dentability, V-s-dentability and V-f-dentability are introduced. It is shown, in particular, that if B is a bounded sequentially complete convex metrizable subset of a locally convex vector space E and V is a neighborhood of zero in E, then the following are equivalent: 1). B is subset V-dentable; 2). B is subset V-s-dentable; 3). B is subset V-f-dentable. It follows from this that for a wide class of locally convex vector spaces E, which strictly contains the class of (BM) spaces (introduced by Elias Saab in 1978), the following is true: every closed bounded subset of E is dentable if and only if every closed bounded subset of E is f-dentable. Also, we get a positive answer to the Saab's question (1978) of whether the subset dentability and the subset s-dentability are the same forthe bounded complete convex metrizable subsets of any l.c.v. space.

scholarly journals Functional envelope of a non-autonomous discrete system

2017 ◽  
Vol 4 (1) ◽  
pp. 98-107
Author(s):  
Ali Barzanouni
Keyword(s):  
Metric Space ◽  
Discrete System ◽  
Compact Metric Space ◽  
Continuous Maps ◽  
Functional Envelope

Abstract Let (X, F = {fn}n =0∞) be a non-autonomous discrete system by a compact metric space X and continuous maps fn : X → X, n = 0, 1, ....We introduce functional envelope (S(X), G = {Gn}n =0∞), of (X, F = {fn}n =0∞), where S(X) is the space of all continuous self maps of X and the map Gn : S(X) → S(X) is defined by Gn(ϕ) = Fn ∘ ϕ, Fn = fn ∘ fn-1 ∘ . . . ∘ f1 ∘ f0. The paper mainly deals with the connection between the properties of a system and the properties of its functional envelope.

scholarly journals Note on limits of simply continuous and cliquish functions

1994 ◽  
Vol 17 (3) ◽  
pp. 447-450 ◽  
Author(s):  
Janina Ewert
Keyword(s):  
Metric Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Baire Property ◽  
Baire Class ◽  
Class 1 ◽  
Pointwise Limit ◽  
Separable Metric Space

The main result of this paper is that any functionfdefined on a perfect Baire space(X,T)with values in a separable metric spaceYis cliquish (has the Baire property) iff it is a uniform (pointwise) limit of sequence{fn:n≥1}of simply continuous functions. This result is obtained by a change of a topology onXand showing that a functionf:(X,T)→Yis cliquish (has the Baire property) iff it is of the Baire class 1 (class 2) with respect to the new topology.

THE BAIRE PROPERTY IN THE REMAINDERS OF SEMITOPOLOGICAL GROUPS

2013 ◽  
Vol 88 (2) ◽  
pp. 301-308 ◽  
Author(s):  
LI-HONG XIE ◽  
SHOU LIN
Keyword(s):  
Positive Answer ◽  
Baire Space ◽  
Baire Property ◽  
Topological Groups ◽  
Dichotomy Theorem ◽  
Semitopological Group

AbstractIt is proved that every remainder of a nonlocally compact semitopological group $G$ is a Baire space if and only if $G$ is not Čech-complete, which improves a dichotomy theorem of topological groups by Arhangel’skiǐ [‘The Baire property in remainders of topological groups and other results’, Comment. Math. Univ. Carolin. 50(2) (2009), 273–279], and also gives a positive answer to a question of Lin and Lin [‘About remainders in compactifications of paratopological groups’, ArXiv: 1106.3836v1 [Math. GN] 20 June 2011]. We also show that for a nonlocally compact rectifiable space $G$ every remainder of $G$ is either Baire, or meagre and Lindelöf.

scholarly journals Uniform Convergence and Transitive Subsets

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Lei Liu ◽  
Shuli Zhao ◽  
Hongliang Liang
Keyword(s):  
Uniform Convergence ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Continuous Maps

Let(X,d)be a metric space and a sequence of continuous mapsfn:X→Xthat converges uniformly to a mapf. We investigate the transitive subsets offnwhether they can be inherited byfor not. We give sufficient conditions such that the limit mapfhas a transitive subset. In particular, we show the transitive subsets offnthat can be inherited byfiffnconverges uniformly strongly tof.

scholarly journals Estimates of topological entropy of continuous maps with applications

2006 ◽  
Vol 2006 ◽  
pp. 1-10 ◽  
Author(s):  
Xiao-Song Yang ◽  
Xiaoming Bai
Keyword(s):  
Topological Entropy ◽  
Simple Proof ◽  
Metric Space ◽  
Simple Theory ◽  
Flow Networks ◽  
Compact Metric Space ◽  
Continuous Maps

We present a simple theory on topological entropy of the continuous maps defined on a compact metric space, and establish some inequalities of topological entropy. As an application of the results of this paper, we give a new simple proof of chaos in the so-calledN-buffer switched flow networks.

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