scholarly journals Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

2015 ◽  
Vol 58 (2) ◽  
pp. 334-349 ◽  
Author(s):  
Andrea Medini
Keyword(s):  
Metrizable Space ◽  
Countable Subset ◽  
Separable Metrizable Space ◽  
Equivalence Result

AbstractWe show that for a coanalytic subspace X of 2ω, the countable dense homogeneity of Xω is equivalent to X being Polish. This strengthens a result of Hruˇs´ak and Zamora Avilés. Then, inspired by results of Hernández-Guti´errez, Hruˇs´ak, and van Mill, using a technique of Medvedev, we construct a non-Polish subspace X of 2ω such that Xω is countable dense homogeneous. This gives the ûrst ZFC answer to a question of Hruˇs´ak and Zamora Avil´es. Furthermore, since our example is consistently analytic, the equivalence result mentioned above is sharp. Our results also answer a question of Medini and Milovich. Finally, we show that if every countable subset of a zero-dimensional separable metrizable space X is included in a Polish subspace of X, then Xω is countable dense homogeneous.

scholarly journals Infinite Powers and Cohen Reals

2018 ◽  
Vol 61 (4) ◽  
pp. 812-821 ◽  
Author(s):  
Andrea Medini ◽  
Jan van Mill ◽  
Lyubomyr Zdomskyy
Keyword(s):  
Open Problem ◽  
Metrizable Space ◽  
Almost Everywhere ◽  
Separable Metrizable Space ◽  
Insight Into

AbstractWe give a consistent example of a zero-dimensional separable metrizable space Z such that every homeomorphism of Zω acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example Z is simply the set of ω1 Cohen reals, viewed as a subspace of 2ω.

Baire sets that are Borelian subspaces

1967 ◽  
Vol 299 (1457) ◽  
pp. 287-290 ◽  
Keyword(s):  
Continuous Mapping ◽  
Metrizable Space ◽  
Closed Sets ◽  
Open Set ◽  
Separable Metrizable Space

The Borelian subspaces that are Baire sets were studied by Knowles & Rogers (1966). These sets are here called BB-sets and are characterized among all Borelian subspaces by the property of being separated from their complements under some continuous mapping into some separable metrizable space. The main properties of the BB-sets are developed by the methods of Frolík (1961). It is shown that every Borelian subspace of a space P is a Baire set in P if each open set of P is a Souslin set derived from the closed sets.

A Hilbert cube compactification of the function space with the compact-open topology

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Atsushi Kogasaka ◽  
Katsuro Sakai
Keyword(s):  
Closed Interval ◽  
Continuous Functions ◽  
Metrizable Space ◽  
Hilbert Cube ◽  
Compact Open Topology ◽  
Locally Compact ◽  
Closed Sets ◽  
Isolated Points ◽  
Compact Case ◽  
Separable Metrizable Space

AbstractLet X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $$ \bar C $$(X) of C(X) such that the pair ($$ \bar C $$(X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification $$ \bar C $$(X) coincides with the space USCCF(X,) of all upper semi-continuous set-valued functions φ: X → = [−∞, ∞] such that each φ(x) is a closed interval, where the topology for USCCF(X, ) is inherited from the Fell hyperspace Cld*F(X × ) of all closed sets in X × .

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.

Uncountable models and infinitary elementary extensions

1973 ◽  
Vol 38 (3) ◽  
pp. 460-470 ◽  
Author(s):  
John Gregory
Keyword(s):  
Countable Subset ◽  
Elementary Extension ◽  
Admissible Set ◽  
Natural Numbers ◽  
Elementary Equivalence ◽  
Countable Structure

Let A be a countable admissible set (as defined in [1], [3]). The language LA consists of all infinitary finite-quantifier formulas (identified with sets, as in [1]) that are elements of A. Notationally, LA = A ∩ Lω1ω. Then LA is a countable subset of Lω1ω, the language of all infinitary finite-quantifier formulas with all conjunctions countable. The set is the set of Lω1ω sentences defined in 2.2 below. The following theorem characterizes those A-Σ1 sets Φ of LA sentences that have uncountable models.Main Theorem (3.1.). If Φ is an A-Σ1set of LA sentences, then the following are equivalent:(a) Φ has an uncountable model,(b) Φ has a model with a proper LA-elementary extension,(c) for every , ⋀Φ → C is not valid.This theorem was announced in [2] and is proved in §§3, 4, 5. Makkai's earlier [4, Theorem 1] implies that, if Φ determines countable structure up to Lω1ω-elementary equivalence, then (a) is equivalent to (c′) for all , ⋀Φ → C is not valid.The requirement in 3.1 that Φ is A-Σ1 is essential when the set ω of all natural numbers is an element of A. For by the example of [2], then there is a set Φ LA sentences such that (b) holds and (a) fails; it is easier to show that, if ω ϵ A, there is a set Φ of LA sentences such that (c) holds and (b) fails.

scholarly journals Remarks on neighborhood star-Lindelöf spaces II

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 875-880
Author(s):  
Yan-Kui Song
Keyword(s):  
Open Cover ◽  
Countable Subset ◽  
Topological Properties ◽  
Pseudocompact Spaces ◽  
The Relationship

A space X is said to be neighborhood star-Lindel?f if for every open cover U of X there exists a countable subset A of X such that for every open O?A, X=St(O,U). In this paper, we continue to investigate the relationship between neighborhood star-Lindel?f spaces and related spaces, and study topological properties of neighborhood star-Lindel?f spaces in the classes of normal and pseudocompact spaces. .

scholarly journals The Metrizability of Spaces whose Diagonals have a Countable Base

1977 ◽  
Vol 20 (4) ◽  
pp. 513-514 ◽  
Author(s):  
John Ginsburg
Keyword(s):  
Metrizable Space ◽  
Countable Base ◽  
Neighborhood Base ◽  
Isolated Points

AbstractIt is shown that the diagonal of X has a countable neighborhood base in X × X if and only if X is a metrizable space whose set of non-isolated points is compact.

An equivalence result for temporal logic

1981 ◽  
Vol 16 (2) ◽  
pp. 53-55
Author(s):  
Stephen C. Dewhurst
Keyword(s):  
Temporal Logic ◽  
Equivalence Result
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