scholarly journals Separate and Joint Properties of some Analogues of Pointwise Discontinuity

2014 ◽  
Vol 58 (1) ◽  
pp. 155-167
Author(s):  
Vasyl’ Nesterenko
Keyword(s):  
Metric Space ◽  
Baire Space ◽  
Residual Subset ◽  
Functions Of Two Variables ◽  
Joint Properties ◽  
Condition C

Abstract We study separate and joint properties of pointwise discontinuity, simple continuity and mild continuity of functions of two variables. In particular, it is shown that for a Baire space X, aBaire space Y which has a countable pseudobase and for a metric space Z, a function ƒ : X×Y → Z is pointwise discontinuous if and only if f satisfies (α, β)-condition and condition (C), and M = {x ∊ X : C(ƒx) = Y } is a residual subset of X. In addition, a characterization of simple continuity for mappings of one and two variables is given

scholarly journals Equi-Cliquishness and The Hahn Property

2016 ◽  
Vol 65 (1) ◽  
pp. 85-92
Author(s):  
Vasyl’ Nesterenko
Keyword(s):  
Metric Space ◽  
Baire Space ◽  
Residual Subset ◽  
Joint Continuity ◽  
Countable Space ◽  
Second Countable Space

Abstract We study the joint continuity of mappings of two variables. In particular, we show that for a Baire space X, a second countable space Y and a metric space Z, a map f : X × Y → Z has the Hahn property (i.e., there is a residual subset A of X such that A × Y ⊆ C(f)) if and only if f is locally equi-cliquish with respect to y and {x ∈ X: fx is continuous} is a residual subset of X.

scholarly journals An Intrinsic Characterization of Five Points in a CAT(0) Space

2020 ◽  
Vol 8 (1) ◽  
pp. 114-165
Author(s):  
Tetsu Toyoda
Keyword(s):  
Metric Space ◽  
Isometric Embedding ◽  
Metric Spaces ◽  
Necessary Conditions ◽  
New Approach ◽  
Intrinsic Characterization ◽  
New Family

AbstractGromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

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 $\textrm {C}(K)$ .

On a theorem of Brian Fisher in the framework of w-distance

2017 ◽  
Vol 33 (2) ◽  
pp. 199-205
Author(s):  
DARKO KOCEV ◽  
◽  
VLADIMIR RAKOCEVIC ◽  
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Complete Metric Space ◽  
Point Theory ◽  
Common Fixed Points ◽  
Complete Metric Spaces ◽  
Condition C ◽  
Relaxing Condition

In 1980. Fisher in [Fisher, B., Results on common fixed points on complete metric spaces, Glasgow Math. J., 21 (1980), 165–167] proved very interesting fixed point result for the pair of maps. In 1996. Kada, Suzuki and Takahashi introduced and studied the concept of w–distance in fixed point theory. In this paper, we generalize Fisher’s result for pair of mappings on metric space to complete metric space with w–distance. The obtained results do not require the continuity of maps, but more relaxing condition (C; k). As a corollary we obtain a result of Chatterjea.

scholarly journals On Almost Continuous Mappings and Baire Spaces

1978 ◽  
Vol 21 (2) ◽  
pp. 183-186 ◽  
Author(s):  
Shwu-Yeng T. Lin ◽  
You-Feng Lin
Keyword(s):  
Topological Space ◽  
Dense Subset ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Continuous Mappings ◽  
Almost Continuous

AbstractIt is proved, in particular, that a topological space X is a Baire space if and only if every real valued function f: X →R is almost continuous on a dense subset of X. In fact, in the above characterization of a Baire space, the range space R of real numbers may be generalized to any second countable, Hausdorfï space that contains infinitely many points.

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.

scholarly journals A characterization of locally connected unicoherent continua

1969 ◽  
Vol 10 (3-4) ◽  
pp. 257-265
Author(s):  
Philip Bacon
Keyword(s):  
Metric Space ◽  
Finite Sequence ◽  
Compact Metric Space

If ε > 0, a subset M of a metric space is said to be ε-connected if for each pair p, q ∈ M there is a finite sequence a0, …, an such that each ai ∈ M, a0 = ρ an = q and the distance from ai−1 to ai is less than ε whenever 0 < i ≦n. It is known [1, p. 117, Satz 1] that a compact metric space is connected if and only if for each ε > 0 it is ε-connected. We present here a proof of an analogous characterization of locally connected unicoherent compacta.

MAXIMAL COMPUTABILITY STRUCTURES

2016 ◽  
Vol 22 (4) ◽  
pp. 445-468 ◽  
Author(s):  
ZVONKO ILJAZOVIĆ ◽  
LUCIJA VALIDŽIĆ
Keyword(s):  
Euclidean Space ◽  
Metric Space ◽  
Dense Sequence

AbstractA computability structure on a metric space is a set of sequences which satisfy certain conditions. Of a particular interest are those computability structures which contain a dense sequence, so called separable computability structures. In this paper we observe maximal computability structures which are more general than separable computability structures and we examine their properties. In particular, we examine maximal computability structures on subspaces of Euclidean space, we give their characterization and we investigate conditions under which a maximal computability structure on such a space is unique. We also give a characterization of separable computability structures on a segment.

A theorem about countable decomposability

1982 ◽  
Vol 91 (3) ◽  
pp. 457-458 ◽  
Author(s):  
Roy O. Davies ◽  
Claude Tricot
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Continuous Functions ◽  
Perfect Set ◽  
Baire Function ◽  
Separable Metric Space

A function f:X → ℝ is countably decomposable (into continuous functions) if the topological space X can be partitioned into countably many sets An with each restriction f│ An continuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric space X (see Corollary 1). We deduce that when X is complete they include all functions possessing the property P defined by D. E. Peek in (3): each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has property P.

scholarly journals Closedness, separation and connectedness in pseudo-quasi-semi metric spaces

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4757-4766
Author(s):  
Tesnim Baran
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Closure Operator ◽  
Topological Category ◽  
The Relationship

In this paper, we give the characterization of closed and strongly closed subsets of an extended pseudo-quasi-semi metric space and show that they induce closure operator. Moreover, we characterize each of Ti, i = 0, 1, 2 and connected extended pseudo-quasi-semi metric spaces and investigate the relationship among them. Finally, we introduce the notion of irreducible objects in a topological category and examine the relationship among each of irreducible, Ti,i = 1,2, and connected extended pseudo-quasi-semi metric spaces.

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