Zero-dimensional CDH spaces with a dense completely metrizable subset

2019 ◽  
Vol 159 (1) ◽  
pp. 164-173
Author(s):  
S. V. Medvedev
Keyword(s):  
Completely Metrizable

Topological Completeness of Function Spaces Arising in the Hausdorff Approximation of Functions

1992 ◽  
Vol 35 (4) ◽  
pp. 439-448 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Approximation Theory ◽  
Metric Space ◽  
Polish Space ◽  
Complete Metric Space ◽  
Approximation Of Functions ◽  
Lower Envelopes ◽  
Constructive Approximation ◽  
Hausdorff Approximation ◽  
Set Of Points ◽  
Completely Metrizable

AbstractLet X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?

Non-Extendability of Bounded Continuous Functions

1980 ◽  
Vol 32 (4) ◽  
pp. 867-879
Author(s):  
Ronnie Levy
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Compact Metric Space ◽  
Bounded Continuous Function ◽  
Bounded Functions ◽  
Bounded Continuous Functions ◽  
Completely Metrizable

If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).

scholarly journals THE MAPPING CLASS GROUP OF A DISK WITH INFINITELY MANY HOLES

2006 ◽  
Vol 15 (01) ◽  
pp. 21-29 ◽  
Author(s):  
PAUL FABEL
Keyword(s):  
Topological Group ◽  
Unit Disk ◽  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Closed Unit Disk ◽  
Completely Metrizable ◽  
Metrizable Topological Group

A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehornoy.

scholarly journals Function spaces of completely metrizable spaces

1993 ◽  
Vol 340 (2) ◽  
pp. 871-883 ◽  
Author(s):  
Jan Baars ◽  
Joost de Groot ◽  
Jan Pelant
Keyword(s):  
Function Spaces ◽  
Metrizable Spaces ◽  
Completely Metrizable

Cell structures and completely metrizable spaces and their mappings

2017 ◽  
Vol 147 (2) ◽  
pp. 181-194 ◽  
Author(s):  
Wojciech Dębski ◽  
E. D. Tymchatyn
Keyword(s):  
Cell Structures ◽  
Metrizable Spaces ◽  
Completely Metrizable

scholarly journals RIGIDITY OF CONTINUOUS QUOTIENTS

2014 ◽  
Vol 15 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Ilijas Farah ◽  
Saharon Shelah
Keyword(s):  
Connected Space ◽  
Proper Forcing Axiom ◽  
Reduced Products ◽  
Proper Forcing ◽  
Metric Structures ◽  
Continuous Fields ◽  
The Axiom Of Choice ◽  
Metrizable Spaces ◽  
Completely Metrizable ◽  
Index Space

We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand–Naimark duality we conclude that the assertion that the Stone–Čech remainder of the half-line has only trivial automorphisms is independent from ZFC (Zermelo-Fraenkel axiomatization of set theory with the Axiom of Choice). Consistency of this statement follows from the Proper Forcing Axiom, and this is the first known example of a connected space with this property.

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