scholarly journals Cardinal estimates involving the weak Lindelöf game

2021 ◽  
Vol 116 (1) ◽  
Author(s):  
Leandro Aurichi ◽  
Angelo Bella ◽  
Santi Spadaro
Keyword(s):  
Compact Space ◽  
Winning Strategy ◽  
Partial Answer ◽  
Large Cardinality ◽  
Countable Space ◽  
First Countable Space

AbstractWe show that if X is a first-countable Urysohn space where player II has a winning strategy in the game $$G^{\omega _1}_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 ω 1 ( O , O D ) (the weak Lindelöf game of length $$\omega _1$$ ω 1 ) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a winning strategy in the game $$G^{\omega _1}_{fin}({\mathcal {O}}, {\mathcal {O}}_D)$$ G fin ω 1 ( O , O D ) , providing some partial answers to it. We finish by constructing an example of a compact space where player II does not have a winning strategy in the weak Lindelöf game of length $$\omega _1$$ ω 1 .

scholarly journals Embedding inverse semigroups of homeomorphisms on closed subsets

1977 ◽  
Vol 18 (2) ◽  
pp. 199-207 ◽  
Author(s):  
Bridget Bos Baird
Keyword(s):  
Topological Space ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Spaces ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Countable Space ◽  
The One ◽  
One Point Compactification ◽  
First Countable Space

All topological spaces here are assumed to be T2. The collection F(Y)of all homeomorphisms whose domains and ranges are closed subsets of a topological space Y is an inverse semigroup under the operation of composition. We are interested in the general problem of getting some information about the subsemigroups of F(Y) whenever Y is a compact metric space. Here, we specifically look at the problem of determining those spaces X with the property that F(X) is isomorphic to a subsemigroup of F(Y). The main result states that if X is any first countable space with an uncountable number of points, then the semigroup F(X) can be embedded into the semigroup F(Y) if and only if either X is compact and Y contains a copy of X, or X is noncompact and locally compact and Y contains a copy of the one-point compactification of X.

scholarly journals The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces

2019 ◽  
Vol 69 (1) ◽  
pp. 185-198
Author(s):  
Fadoua Chigr ◽  
Frédéric Mynard
Keyword(s):  
Space Structures ◽  
Theoretic Approach ◽  
Hausdorff Spaces ◽  
Sequential Space ◽  
Countable Space ◽  
The Stability ◽  
First Countable Space ◽  
Algebraic Proofs ◽  
Non Hausdorff Spaces

AbstractThis article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out convergence-theoretic characterizations of the notions of weak base, weakly first-countable space, semi-metrizable space, and symmetrizable spaces. With the help of the already established similar characterizations of the notions of Frchet-Ursyohn, sequential, and accessibility spaces, we give a simple algebraic proof of a classical result regarding when a symmetrizable (respectively, weakly first-countable, respectively sequential) space is semi-metrizable (respectively first-countable, respectively Fréchet) that clarifies the situation for non-Hausdorff spaces. Using additionally known results on the commutation of the topologizer with product, we obtain simple algebraic proofs of various results of Y. Tanaka on the stability under product of symmetrizability and weak first-countability, and we obtain the same way a new characterization of spaces whose product with every metrizable topology is weakly first-countable, respectively symmetrizable.

scholarly journals What is a first countable space?

2006 ◽  
Vol 153 (18) ◽  
pp. 3420-3429 ◽  
Author(s):  
Gonçalo Gutierres
Keyword(s):  
Countable Space ◽  
First Countable Space

scholarly journals Different versions of a first countable space without choice

2009 ◽  
Vol 156 (12) ◽  
pp. 2000-2004
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis
Keyword(s):  
Countable Space ◽  
First Countable Space

scholarly journals A better framework for first countable spaces

2003 ◽  
Vol 4 (2) ◽  
pp. 289
Author(s):  
Gerhard Preuss
Keyword(s):  
Topological Space ◽  
Uniform Space ◽  
Topological Spaces ◽  
Convergence Space ◽  
Convergence Spaces ◽  
Natural Function ◽  
Topological Construct ◽  
Countable Space ◽  
Filter Space ◽  
First Countable Space

<p>In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. Every semiuniform convergence space (e.g. symmetric topological space, uniform space, filter space, etc.) has an underlying first countable space. Several applications of first countability in a broader context than the usual one of topological spaces are studied.</p>

Separating Points and Coloring Principles

1984 ◽  
Vol 27 (4) ◽  
pp. 398-404
Author(s):  
W. Stephen Watson
Keyword(s):  
Binary Tree ◽  
Weak Version ◽  
Countable Space ◽  
First Countable Space ◽  
Discrete Family

AbstractIn the mid 1970's, Shelah formulated a weak version of ◊. This axiom Φ is a prediction principle for colorings of the binary tree of height ω1. Shelah and Devlin showed that Φ is equivalent to 2ℵ0 < 2ℵ1.In this paper, we formulate Φp, a "Φ for partial colorings", show that both ◊* and Fleissner's “◊ for stationary systems” imply Φp, that ◊ does not imply Φp and that Φp does not imply CH.We show that Φp implies that, in a normal first countable space, a discrete family of points of cardinality ℵ1 is separated.

scholarly journals On Levine′s Decomposition of Continuity

1978 ◽  
Vol 21 (4) ◽  
pp. 477-481 ◽  
Author(s):  
David Alon Rose
Keyword(s):  
Compact Space ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Locally Compact ◽  
Weakly Continuous ◽  
Countable Space ◽  
Second Countable Space ◽  
Graph Function ◽  
Almost Continuous

AbstractA strong version of Levine′s decomposition of continuity leads to the result that a closed graph weakly continuous function into a rim-compact space is continuous. This result implies a closed graph theorem: every almost continuous closed graph function into a strongly locally compact space is continuous. An open problem of Shwu-Yeng T. Lin and Y.-F. Lin asks if every almost continuous closed graph function from a Baire space to a second countable space is necessarily continuous. This question is answered in the negative by an example.

scholarly journals Compact complement topologies and k-spaces

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2061-2071
Author(s):  
K. Keremedis ◽  
C. Özel ◽  
A. Piękosz ◽  
Shumrani Al ◽  
E. Wajch
Keyword(s):  
Hausdorff Space ◽  
Multiple Choice ◽  
Metrizable Space ◽  
Paper Properties ◽  
Countable Space ◽  
Sorgenfrey Line ◽  
First Countable Space ◽  
Infinite Set

Let (X,?) be a Hausdorff space, where X is an infinite set. The compact complement topology ?* on X is defined by: ?* = {0}?{X\M:M is compact in (X,?)}. In this paper, properties of the space (X,?*) are studied in ZF and applied to a characterization of k-spaces, to the Sorgenfrey line, to some statements independent of ZF, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Between other results, it is proved that the axiom of countable multiple choice (CMC) is equivalent with each of the following two sentences: (i) every Hausdorff first-countable space is a k-space, (ii) every metrizable space is a k-space. A ZF-example of a countable metrizable space whose compact complement topology is not first-countable is given.

scholarly journals Some Results in Neutrosophic Soft Topology Concerning Neutrosophic Soft ∗ b Open Sets

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Arif Mehmood ◽  
Saleem Abdullah ◽  
Mohammed M. Al-Shomrani ◽  
Muhammad Imran Khan ◽  
Orawit Thinnukool
Keyword(s):  
Compact Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Soft Topology ◽  
Open Set ◽  
Topological Features ◽  
Separation Axioms ◽  
Countable Space ◽  
Countably Compact

In this article, new generalised neutrosophic soft open known as neutrosophic soft ∗ b open set is introduced in neutrosophic soft topological spaces. Neutrosophic soft ∗ b open set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. Then, with the application of this new definition, some soft neutrosophical separation axioms, countability theorems, and countable space can be Hausdorff space under the subjection of neutrosophic soft sequence which is convergent, the cardinality of neutrosophic soft countable space, engagement of neutrosophic soft countable and uncountable spaces, neutrosophic soft topological features of the various spaces, soft neutrosophical continuity, the product of different soft neutrosophical spaces, and neutrosophic soft countably compact that has the characteristics of Bolzano Weierstrass Property (BVP) are studied. In addition to this, BVP shifting from one space to another through neutrosophic soft continuous functions, neutrosophic soft sequence convergence, and its marriage with neutrosophic soft compact space, sequentially compactness are addressed.

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