Metric Spaces: Continuity, Topological Notions, Compact Sets

1998 ◽  
pp. 75-96
Author(s):  
Jürgen Jost
Keyword(s):  
Metric Spaces ◽  
Compact Sets

scholarly journals Isometries of spaces of unbounded continuous functions

2001 ◽  
Vol 63 (3) ◽  
pp. 475-484
Author(s):  
Jesús Araujo ◽  
Krzysztof Jarosz
Keyword(s):  
Banach Spaces ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Compact Sets ◽  
Surjective Isometry ◽  
Bounded Continuous Functions ◽  
Stone Theorem

By the classical Banach-Stone Theorem any surjective isometry between Banach spaces of bounded continuous functions defined on compact sets is given by a homeomorphism of the domains. We prove that the same description applies to isometries of metric spaces of unbounded continuous functions defined on non compact topological spaces.

Domain representations of spaces of compact subsets

2010 ◽  
Vol 20 (2) ◽  
pp. 107-126 ◽  
Author(s):  
ULRICH BERGER ◽  
JENS BLANCK ◽  
PETTER KRISTIAN KØBER
Keyword(s):  
Metric Spaces ◽  
Topological Properties ◽  
Compact Sets ◽  
Natural Choice ◽  
Compact Subsets

We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domain representations with a natural choice of morphisms. We study the topological properties of the space of representable compact sets and isolate conditions under which all compact subsets of X are representable. Special attention is paid to admissible representations and representations of metric spaces.

Metric Spaces

2021 ◽  
pp. 103-125
Author(s):  
James Davidson
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Distance Measure ◽  
Function Spaces ◽  
Important Case ◽  
General Context ◽  
Compact Sets ◽  
Definition Of ◽  
Metric Distance

This chapter introduces and illustrates the concept of a metric (distance measure), and the definition of a metric space. Open, closed, and compact sets are discussed in a general context, and the concepts of separability and completeness introduced. It goes on to look at mappings on metric spaces, examines the important case of function spaces, and treats the Arzelà–Ascoli theorem.

scholarly journals Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Evgeniy Pustylnik ◽  
Simeon Reich ◽  
Alexander J. Zaslavski
Keyword(s):  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Compact Sets

Borel and projective sets from the point of view of compact sets

1983 ◽  
Vol 94 (3) ◽  
pp. 399-409 ◽  
Author(s):  
Kenneth Kunen ◽  
Arnold W. Miller
Keyword(s):  
Metric Spaces ◽  
Polish Space ◽  
Point Of View ◽  
Alternative Proof ◽  
Compact Sets ◽  
Borel Set ◽  
Compact Metric Spaces ◽  
Borel Complexity ◽  
Continuous Surjection

In this paper we prove several results concerning the complexity of a set relative to compact sets. We prove that for any Polish space X and Borel set B ⊆ X, if B is not , then there exists a compact zero-dimensional P ⊆ X such that p ∩ X is not . We also show that it is consistent with ZFC that, for any A ⊆ ωω, if for all compact K ⊆ ωωA ∩ K is , then A is . This generalizes to in place of assuming the consistency of some hypotheses involving determinacy. We give an alternative proof of the following theorem of Saint-Raymond. Suppose X and Y are compact metric spaces and f is a continuous surjection of X onto Y. Then, for any A ⊆ Y, A is in Y iff f−1(A) is in X. The non-trivial part of this result is to show that taking pre-images cannot reduce the Borel complexity of a set. The techniques we use are the definability of forcing and Wadge games.

scholarly journals New examples of subsets of c with the FPP and stability of the FPP in hyperconvex spaces

2021 ◽  
Vol 23 (3) ◽  
Author(s):  
Rafael Espínola-García ◽  
María Japón ◽  
Daniel Souza
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Point Property ◽  
Compact Sets ◽  
Weakly Compact ◽  
Weakly Compact Sets ◽  
The One

AbstractThe purpose of this work is two-fold. On the one side, we focus on the space of real convergent sequences c where we study non-weakly compact sets with the fixed point property. Our approach brings a positive answer to a recent question raised by Gallagher et al. in (J Math Anal Appl 431(1):471–481, 2015). On the other side, we introduce a new metric structure closely related to the notion of relative uniform normal structure, for which we show that it implies the fixed point property under adequate conditions. This will provide some stability fixed point results in the context of hyperconvex metric spaces. As a particular case, we will prove that the set $$M=[-1,1]^\mathbb {N}$$ M = [ - 1 , 1 ] N has the fixed point property for d-nonexpansive mappings where $$d(\cdot ,\cdot )$$ d ( · , · ) is a metric verifying certain restrictions. Applications to some Nakano-type norms are also given.

scholarly journals REMARKS ON DIMENSIONS OF CARTESIAN PRODUCT SETS

Fractals ◽  
2016 ◽  
Vol 24 (03) ◽  
pp. 1650031 ◽  
Author(s):  
CHUN WEI ◽  
SHENGYOU WEN ◽  
ZHIXIONG WEN
Keyword(s):  
Metric Spaces ◽  
Cartesian Product ◽  
Product Formula ◽  
Compact Sets ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Cartesian Product Sets ◽  
Product Sets

Given metric spaces [Formula: see text] and [Formula: see text], it is well known that [Formula: see text] [Formula: see text] and [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] denote the Hausdorff, packing, lower box-counting, and upper box-counting dimension of [Formula: see text], respectively. In this paper, we shall provide examples of compact sets showing that the dimension of the product [Formula: see text] may attain any of the values permitted by the above inequalities. The proof will be based on a study on dimension of products of sets defined by digit restrictions.

scholarly journals A thermodynamic definition of topological pressure for non-compact sets

2010 ◽  
Vol 31 (2) ◽  
pp. 527-547 ◽  
Author(s):  
DANIEL J. THOMPSON
Keyword(s):  
Lyapunov Exponent ◽  
Variational Principle ◽  
Equilibrium State ◽  
Metric Space ◽  
Level Sets ◽  
Metric Spaces ◽  
Topological Pressure ◽  
Compact Sets ◽  
Alternative Definition ◽  
Definition Of

AbstractWe give a new definition of topological pressure for arbitrary (non-compact, non-invariant) Borel subsets of metric spaces. This new quantity is defined via a suitable variational principle, leading to an alternative definition of an equilibrium state. We study the properties of this new quantity and compare it with existing notions of topological pressure. We are particularly interested in the situation when the ambient metric space is assumed to be compact. We motivate our definition by applying it to some interesting examples, including the level sets of the pointwise Lyapunov exponent for the Manneville–Pomeau family of maps.

scholarly journals Banach-Like Distances and Metric Spaces of Compact Sets

2015 ◽  
Vol 8 (1) ◽  
pp. 19-66 ◽  
Author(s):  
A. C. G. Mennucci ◽  
A. Duci
Keyword(s):  
Metric Spaces ◽  
Compact Sets
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