Correction to "A Note on Completely Metrizable Spaces"

1987 ◽  
Vol 100 (1) ◽  
pp. 204
Author(s):  
E. Michael
Keyword(s):  
Metrizable Spaces ◽  
Completely Metrizable

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.

scholarly journals Locally n-Connected Compacta and UVn-Maps

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
V. Valov
Keyword(s):  
Inverse System ◽  
Limit Space ◽  
A Cell ◽  
Metrizable Spaces ◽  
Completely Metrizable ◽  
Spectral Characterizations

Abstract We provide a machinery for transferring some properties of metrizable ANR-spaces to metrizable LCn-spaces. As a result, we show that for completely metrizable spaces the properties ALCn, LCn and WLCn coincide to each other. We also provide the following spectral characterizations of ALCn and celllike compacta: A compactum X is ALCn if and only if X is the limit space of a σ-complete inverse system S = {Xα , pβ α , α < β < τ} consisting of compact metrizable LCn-spaces Xα such that all bonding projections pβα, as a well all limit projections pα, are UVn-maps. A compactum X is a cell-like (resp., UVn) space if and only if X is the limit space of a σ-complete inverse system consisting of cell-like (resp., UVn) metrizable compacta.

scholarly journals GAMES AND HEREDITARY BAIRENESS IN HYPERSPACES AND SPACES OF PROBABILITY MEASURES

2020 ◽  
pp. 1-18
Author(s):  
Mikołaj Krupski
Keyword(s):  
Winning Strategy ◽  
Metrizable Space ◽  
Baire Property ◽  
Probability Measures ◽  
Game Theoretic ◽  
Nonempty Compact ◽  
Borel Probability ◽  
Metrizable Spaces ◽  
Completely Metrizable ◽  
Separable Metrizable Space

We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$ , we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$ . Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$ -filter ${\mathcal{F}}$ and prove that it is equivalent to $K({\mathcal{F}})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_{r}(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$ , which is not completely metrizable with $P_{r}(X)$ hereditarily Baire. As far as we know, this is the first example of this kind.

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