The pseudocompactness of [0.1] is equivalent to the uniform continuity theorem

2007 ◽  
Vol 72 (4) ◽  
pp. 1379-1384 ◽  
Author(s):  
Douglas Bridges ◽  
Hannes Diener
Keyword(s):  
Metric Space ◽  
Continuous Mapping ◽  
Uniform Continuity ◽  
Compact Metric Space ◽  
Continuity Theorem ◽  
Continuous Mappings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

AbstractWe prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into ℝ is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.

scholarly journals Semi-smooth Points in Some Classical Function Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Spaces Of Continuous Functions ◽  
Classical Function ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

scholarly journals Necessary and sufficient fixed point critera involving attractors

1993 ◽  
Vol 48 (1) ◽  
pp. 109-116
Author(s):  
Jacek Jachymski
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Complete Metric Space ◽  
Positive Real ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Let f be a continuous self-map on a complete metric space X and p ∈ X. Let c be a positive real. Equivalent conditions are given for the singleton {p} to be an attractor of a set of c−fixed points of f. We also establish equivalent conditions for the existence of a contractive fixed point of f. These results subsume a body of fixed point theorems.

scholarly journals Random products of maps synchronizing on average

2020 ◽  
Vol 22 (08) ◽  
pp. 1950086
Author(s):  
Edgar Matias ◽  
Ítalo Melo
Keyword(s):  
Metric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Random Product ◽  
Random Products ◽  
Necessary And Sufficient

We present a necessary and sufficient condition for a random product of maps on a compact metric space to be (strongly) synchronizing on average.

On topological entropy of maps

1995 ◽  
Vol 15 (3) ◽  
pp. 557-568 ◽  
Author(s):  
Mike Hurley
Keyword(s):  
Topological Entropy ◽  
Metric Space ◽  
Continuous Mapping ◽  
Inverse Image ◽  
Compact Metric Space ◽  
Image Entropy ◽  
Definition Of

AbstractWe introduce an ‘entropy’ hm(f) for a continuous mapping of a compact metric space to itself which is denned in terms of (n, ∈)-separated subsets of inverse images of individual points. This invariant is compared with the inverse-image entropy h_(f) introduced recently by Langevin and Walczak. The two main results are: (1) the inequality hm(f) ≤ h(f) ≤ hm(f) + h_(f) relating hm, h_ and the topological entropy h(f); (2) if pseudo-orbits are used in place of orbits in the definition of hm then the quantity that results is equal to the topological entropy. We actually establish an inequality that at least formally is slightly stronger than (1) by defining a variant of h_ which we call hi; it is trivial to show that hi ≤ h_, and we show that h ≤ hi + hm, from which (1) follows.

LIMITS OF HOMOMORPHISMS WITH FINITE-DIMENSIONAL RANGE

2005 ◽  
Vol 16 (07) ◽  
pp. 807-821 ◽  
Author(s):  
SHANWEN HU ◽  
HUAXIN LIN ◽  
YIFENG XUE
Keyword(s):  
Metric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
C Algebra ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Unital Simple ◽  
Dimensional Range

Let X be a compact metric space and A be a unital simple C*-algebra with TR (A)=0. Suppose that ϕ : C(X) → A is a unital monomorphism. We study the problem when ϕ can be approximated by homomorphisms with finite-dimensional range. We give a K-theoretical necessary and sufficient condition for ϕ being approximated by homomorphisms with finite-dimensional range.

scholarly journals Fuzzy Continuous Mappings in Fuzzy Normed Linear Spaces

2015 ◽  
Vol 10 (6) ◽  
pp. 74 ◽  
Author(s):  
Sorin Nadaban
Keyword(s):  
Continuous Mapping ◽  
Uniform Boundedness ◽  
Uniform Continuity ◽  
Linear Operators ◽  
Closed Graph Theorem ◽  
Linear Spaces ◽  
Open Mapping Theorem ◽  
Normed Linear Spaces ◽  
Continuous Mappings ◽  
Uniform Boundedness Principle

In this paper we continue the study of fuzzy continuous mappings in fuzzy normed linear spaces initiated by T. Bag and S.K. Samanta, as well as by I. Sadeqi and F.S. Kia, in a more general settings. Firstly, we introduce the notion of uniformly fuzzy continuous mapping and we establish the uniform continuity theorem in fuzzy settings. Furthermore, the concept of fuzzy Lipschitzian mapping is introduced and a fuzzy version for Banach’s contraction principle is obtained. Finally, a special attention is given to various characterizations of fuzzy continuous linear operators. Based on our results, classical principles of functional analysis (such as the uniform boundedness principle, the open mapping theorem and the closed graph theorem) can be extended in a more general fuzzy context.

scholarly journals The chain recurrent set, attractors, and explosions

1985 ◽  
Vol 5 (3) ◽  
pp. 321-327 ◽  
Author(s):  
Louis Block ◽  
John E. Franke
Keyword(s):  
Compact Space ◽  
Metric Space ◽  
Periodic Points ◽  
Limit Set ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Special Case ◽  
Equivalent Conditions ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractCharles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x. In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.

Sequentially continuous linear mappings in constructive analysis

1998 ◽  
Vol 63 (2) ◽  
pp. 579-583 ◽  
Author(s):  
Douglas Bridges ◽  
Ray Mines
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Continuous Mapping ◽  
Constructive Mathematics ◽  
Closed Graph Theorem ◽  
Inverse Mapping ◽  
Inverse Mapping Theorem ◽  
Continuous Mappings ◽  
Sequential Continuity ◽  
Separable Metric Space

A mapping u: X → Y between metric spaces is sequentially continuous if for each sequence (xn) converging to x ∈ X, (u(xn)) converges to u(x). It is well known in classical mathematics that a sequentially continuous mapping between metric spaces is continuous; but, as all proofs of this result involve the law of excluded middle, there appears to be a constructive distinction between sequential continuity and continuity. Although this distinction is worth exploring in its own right, there is another reason why sequential continuity is interesting to the constructive mathematician: Ishihara [8] has a version of Banach's inverse mapping theorem in functional analysis that involves the sequential continuity, rather than continuity, of the linear mappings; if this result could be upgraded by deleting the word “sequential”, then we could prove constructively the standard versions of the inverse mapping theorem and the closed graph theorem.Troelstra [9] showed that in Brouwer's intuitionistic mathematics (INT) a sequentially continuous mapping on a separable metric space is continuous. On the other hand, Ishihara [6, 7] proved constructively that the continuity of sequentially continuous mappings on a separable metric space is equivalent to a certain boundedness principle for subsets of ℕ; in the same paper, he showed that the latter principle holds within the recursive constructive mathematics (RUSS) of the Markov School. Since it is not known whether that principle holds within Bishop's constructive mathematics (BISH), of which INT and RUSS are models and which can be regarded as the constructive core of mathematics, the exploration of sequential continuity within BISH holds some interest.

scholarly journals Coincidence and common fixed point theorems in compact Hausdorff spaces

2005 ◽  
Vol 2005 (6) ◽  
pp. 845-853 ◽  
Author(s):  
Zeqing Liu ◽  
Haiyan Gao ◽  
Shin Min Kang ◽  
Yong Soo Kim
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Continuous Mapping ◽  
Common Fixed Points ◽  
Hausdorff Spaces ◽  
Continuous Mappings ◽  
Common Fixed Point Theorems ◽  
Equivalent Conditions

The existence of coincidence and fixed points for continuous mappings in compact Hausdorff spaces is established. Some equivalent conditions of the existence of fixed and common fixed points for any continuous mapping and a pair of mappings in compact Hausdorff spaces are given, respectively. Our results extend, improve, and unify the corresponding results due to Jungck, Liu, and Singh and Rao.

scholarly journals On topological equivalence of R-uniformly continuous fuzzy metric spaces

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4523-4531
Author(s):  
Changqing Li ◽  
Yanlan Zhang ◽  
Jing Zhang
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Uniform Continuity ◽  
Fuzzy Metric Space ◽  
Topological Equivalence ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Necessary And Sufficient ◽  
Uniformly Continuous

The notion of uniform continuity in fuzzy metric spaces was first introduced by George and Veeramani in 1995. Later, Gregori et al. gave some contributions to the theory. As a consequence of the study, we introduce the notion of RUC fuzzy metric space. Also, necessary and sufficient conditions for a fuzzy metric space to be an RUC fuzzy metric space are studied. In addition, several examples are given.

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