Cardinal estimates involving the weak Lindelöf game

We 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 .

Locality and evolution to equilibrium

Quantum statistics and non-locality are deeply rooted in quantum mechanics and go beyond our intuition reflected in classical physics. Quantum statistics can be derived using statistical methods for indistinguishable particles - particles of quantum mechanics. Violation of strong locality - colloquially called the ghostly action at a distance - is one of the most amazing properties of nature derived from quantum mechanics. An intriguing question is whether the non-local evolution of indistinguishable particles is needed to reach the equilibrium state given by quantum statistics. Motivated by the above and similar questions, we developed a simple framework that allows us to follow space-time evolution of assembly of particles. It is based on a discrete-time Markov chain on countable space for indistinguishable particles. We summarise well-known and introduced new constraints on the transition matrix that grant space-time symmetries, locality of particle-transport, strong locality, and equilibrium state. Then, within the framework, several important cases are considered. First, we show that the simplest transition matrix leads to equilibrium but violates particle transport and strong localities. Furthermore, we construct a simple matrix that leads to equilibrium obeying particle-transport locality and violating strong locality. This resembles the properties of quantum mechanics. Finally, we demonstrate that it is also possible to reach equilibrium by obeying both particle-transport and strong localities. Thus, within this framework, the violation of a strong locality is not needed to reach the equilibrium of indistinguishable particles. However, to obey strong locality, a complex structure of the transition matrix is needed. In addition, we comment on distinguishable particles and, in particular, show that their evolution seen by an observer blind to particle differences may look like the evolution of indistinguishable particles with the properties of quantum mechanics. We hope that this work may help to study the relation between symmetries, localities and the evolution to equilibrium for indistinguishable and distinguishable particles.

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

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.

Continuous Probability Distributions over Second-Countable Spaces are Perfectly Supported

We prove that every Borel probability measure over an arbitrary second-countable space vanishing at any singletons has support being a perfect set and being included in some co-countable perfect set. Thus the support of a continuous probability distribution over a second-countable space turns out to admit a richer structure.

Covering properties of Cp(X) and Ck(X)

Let X be a Tychonoff space. We survey some classic and recent results that characterize the topology or cardinality of X when Cp (X) or Ck (X) is covered by certain families of sets (sequences, resolutions, closure-preserving coverings, compact coverings ordered by a second countable space) which swallow or not some classes of sets (compact sets, functionally bounded sets, pointwise bounded sets) in C(X).

An ergodic algorithm for generating knots with a prescribed injectivity radius

The first provably ergodic algorithm for sampling the space of thick equilateral knots off-lattice, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections. It is an off-lattice generalization of the pivot algorithm. This move to an off-lattice model provides a huge improvement in power and efficacy in that samples can have arbitrary values for parameters such as the thickness constraint, bending angle, and torsion, while the lattice forces these parameters into a small number of specific values. This benefit requires working in a manifold rather than a finite or countable space, which forces the use of more novel methods in Markov–Chain theory. To prove the validity of the algorithm, we describe a method for turning any knot into the regular planar polygon using only thickness non-decreasing moves. This approach ensures that the algorithm has a positive probability of connecting any two knots with the required thickness constraint which is used to show that the algorithm is ergodic. This ergodic sampling allows for a statistically valid method for estimating probability distributions of arbitrary functions on the space of thick knots.

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

This 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.

Compact complement topologies and k-spaces

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.