Metrizable Topological Space Research Articles

Quantum mechanics on a p-adic Hilbert space: Foundations and prospects

We review some recent results on the mathematical foundations of a quantum theory over a scalar field that is a quadratic extension of the non-Archimedean field of [Formula: see text]-adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, [Formula: see text]-adic string, Class. Quantum Grav. 4 (1987) L83–L87] — that space, below a suitably small scale, does not behave as a continuum and, accordingly, should be modeled as a totally disconnected metrizable topological space, ruled by a metric satisfying the strong triangle inequality. The first step of our construction is a suitable definition of a [Formula: see text]-adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a [Formula: see text]-adic Hilbert space — we consider an algebraic definition of physical states in [Formula: see text]-adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a [Formula: see text]-adic counterpart of the POVMs associated with a standard quantum system over the complex numbers. Interestingly, it turns out that the typical convex geometry of the space of states of a standard quantum system is replaced, in the [Formula: see text]-adic setting, with an affine geometry; therefore, a symmetry transformation of a [Formula: see text]-adic quantum system may be defined as a map preserving this affine geometry. We argue that, as a consequence, the group of all symmetry transformations of a [Formula: see text]-adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers.

A simple characterization of the existence of upper semicontinuous order-preserving functions

We introduce an upper semicontinuity condition concerning a not necessarily total preorder on a topological space, namely strong upper semicontinuity, and in this way we extend to the nontotal case the famous Rader’s theorem, which guarantees the existence of an upper semicontinuous order-preserving function for an upper semicontinuous total preorder on a second countable topological space. We show that Rader’s theorem is not generalizable if we adopt weaker upper semicontinuity conditions already introduced in the literature. We characterize the existence of an upper semicontinuous order-preserving function for all strongly upper semicontinuous preorders on a metrizable topological space.

Chaotic behaviour of countable products of homeomorphism groups

ABSTRACT Relationships between a chaotic behaviour and closely related properties of topological transitivity, sensitivity to initial conditions, density of closed orbits of homeomorphism groups and their countable products are investigated. We provide a large number of new examples of chaotic groups of homeomorphisms of countable products of various metrizable topological spaces, including infinite-dimensional topological manifolds, whose factors can be as noncompact surfaces, so triangulable closed manifolds of an arbitrary dimension.

New Results on the Aggregation of Norms

It is a natural question if a Cartesian product of objects produces an object of the same type. For example, it is well known that a countable Cartesian product of metrizable topological spaces is metrizable. Related to this question, Borsík and Doboš characterized those functions that allow obtaining a metric in the Cartesian product of metric spaces by means of the aggregation of the metrics of each factor space. This question was also studied for norms by Herburt and Moszyńska. This aggregation procedure can be modified in order to construct a metric or a norm on a certain set by means of a family of metrics or norms, respectively. In this paper, we characterize the functions that allow merging an arbitrary collection of (asymmetric) norms defined over a vector space into a single norm (aggregation on sets). We see that these functions are different from those that allow the construction of a norm in a Cartesian product (aggregation on products). Moreover, we study a related topological problem that was considered in the context of metric spaces by Borsík and Doboš. Concretely, we analyze under which conditions the aggregated norm is compatible with the product topology or the supremum topology in each case.

An isomorphism of the Wallman and Čech-Stone compactifications

For a metrizable topological space X it is well known that in general the Čech-Stone compactification β(X) or the Wallman compactification W(X) are not metrizable. To remedy this fact one can alternatively associate a point-set distance to the metric, a so called approach distance. It is known that in this setting both a Čech-Stone compactification β*(X) and a Wallman compactification W*(X) can be constructed in such a way that their approach distances induce the original approach distance of the metric on X [23], [24]. The main goal in this paper is to formulate necessary and sufficient conditions for an approach space X such that the Čech-Stone compactification β*(X) and the Wallman compactification W*(X) are isomorphic, thus answering a question first raised in [24]. The first clue to reach this goal is to settle a question left open in [11], to formulate sufficient conditions for a compact approach space to be normal. In particular the result shows that the Čech-Stone compactification β*(X) of a uniform T 2 space, is always normal. We prove that the Wallman compactification W*(X) is normal if and only if X is normal, and we produce an example showing that, unlike for topological spaces, in the approach setting normality of X is not sufficient for β*(X) and W*(X) to be isomorphic. We introduce a strengthening of the regularity condition on X, which we call ideal-regularity, and in our main theorem we conclude that X is ideal-regular, normal and T 1 if and only if X is a uniform T 1 approach space with β*(X) and W*(X) isomorphic. Classical topological results are recovered and implications for (quasi-)metric spaces are investigated.

On Orthogonally Additive Functions With Big Graphs

Abstract Let E be a separable real inner product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.

AUTOMATIC CONTINUITY OF ALMOST MULTIPLICATIVE LINEAR FUNCTIONALS ON FRÉCHET ALGEBRAS

Abstract. A linear functional T on a Fr´echet algebra (A,(p n )) is calledalmost multiplicative with respect to the sequence (p n ), if there existse≥ 0 such that |Tab− TaTb| ≤ ep n (a)p n (b) for all n∈ N and for everya,b∈ A.We show that an almost multiplicative linear functional on a Fr´echetalgebra is either multiplicative or it is continuous, and hence every almostmultiplicative linear functional on a functionally continuous Fr´echet al-gebra is continuous. 1. IntroductionLet A be an algebra over the complex field. A subset V of A is balanced ifλV ⊆ V for all scalars λ such that |λ| ≤ 1. An algebra A is a Fr´echet algebraif it is a complete metrizable topological linear space and has a neighbourhoodbasis (V n ) of zero such that V n is absolutelyconvex (convex and balanced) andV n is idempotent, i.e., V n V n ⊆ V n for all n ∈ N. The topology of a Fr´echetalgebra A can be generated by a sequence (p n ) of separating submultiplicativeseminorms, i.e., p n (xy) ≤ p n (x)p n (y) for all n ∈ N and x,y ∈ A, such thatp

Fuzzy pseudo-norms and fuzzy F-spaces

In the present paper we firstly introduce the notion of fuzzy pseudo-norm, then we extend, improve and complete the results obtained by T. Bag and S.K. Samanta for fuzzy norms, in the fuzzy pseudo-norms context. Lastly, we introduce and discuss the notions of fuzzy F-norm and fuzzy F-space. By means of several auxiliary results, we obtain a characterization of metrizable topological linear spaces in terms of fuzzy F-norm.

On the Proof of the Existence of Undominated Strategies in Normal Form Games

In the game theory literature, there are two versions of the proof of the well-known fact that in a normal form game of n persons with compact spaces of strategies and continuous utility functions, the sets of undominated strategies are nonempty. The older one, stated in the first edition of the well-known book by Herve Moulin, depends on certain, relatively nontrivial results from measure theory, metric topology, and mathematical analysis. The proof is valid only for metrizable topological spaces. The second, revised edition of the same book contains a simplified proof, which is, however, incorrect. The author implicitly assumes that any linearly ordered set contains a cofinal subsequence, which is certainly not true. In this paper we correct, simplify, and generalize the second proof of Moulin by its reformulation in terms of topological convergence of nets. This modified technique also yields a slightly better result than is stated in the original. The assertion now holds for almost compact spaces. The argument used is elementary and easily understandable to non-experts.

CLASSIFICATION OF CERTAIN CONTINUOUS FIELDS OF KIRCHBERG ALGEBRAS

We show that the K-theory cosheaf is a complete invariant for separable continuous fields with vanishing boundary maps over a finite-dimensional compact metrizable topological space whose fibers are stable Kirchberg algebras with rational K-theory groups satisfying the universal coefficient theorem. We provide a range result for fields in this class with finite-dimensional K-theory. There are versions of both results for unital continuous fields.

Bergeʼs maximum theorem for noncompact image sets

This note generalizes Bergeʼs maximum theorem to noncompact image sets. It also clarifies the results from Feinberg, Kasyanov and Zadoianchuk (2013) [7] on the extension to noncompact image sets of another Bergeʼs theorem, that states semi-continuity of value functions. Here we explain that the notion of a K-inf-compact function introduced there is applicable to metrizable topological spaces and to more general compactly generated topological spaces. For Hausdorff topological spaces we introduce the notion of a KN-inf-compact function (N stands for “nets” in K-inf-compactness), which coincides with K-inf-compactness for compactly generated and, in particular, for metrizable topological spaces.

Metrizable and 2-Metrizable Topological Spaces

In this paper we introduce (∈ − 2)-ball centered at each point in 2-metric topological space (X,d). Theorems on the normal, regular and Hausdorff topological spaces in 2-metrizable topological space are presented. We show that every metrizable topological spaces are coarser than 2-metrizable topological space, and then we conclude that each manifold is a 2-metric topological space.

Topological dynamical systems associated to II 1-factors

If N ⊂ R ω is a separable II 1-factor, the space H om ( N , R ω ) of unitary equivalence classes of unital ⁎-homomorphisms N → R ω is shown to have a surprisingly rich structure. If N is not hyperfinite, H om ( N , R ω ) is an infinite-dimensional, complete, metrizable topological space with convex-like structure, and the outer automorphism group Out ( N ) acts on it by “affine” homeomorphisms. (If N ≅ R , then H om ( N , R ω ) is just a point.) Property (T) is reflected in the extreme points – theyʼre discrete in this case. For certain free products N = Σ ⁎ R , every countable group acts nontrivially on H om ( N , R ω ) , and we show the extreme points are not discrete for these examples. Finally, we prove that the dynamical systems associated to free group factors are isomorphic.

Analytic equivalence relations and bi-embeddability

AbstractLouveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of ) is far from complete (see [5, 2]).In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of , but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids.

Unique ergodicity of horospheric foliations revisited

With each rational function on the Riemann sphere, Lyubich–Minsky construction (1997) associates an abstract topological space called the quotient hyperbolic lamination. The latter space carries the so-called vertical geodesic flow with Anosov property. Its unstable foliation is what we call the quotient horospheric lamination. We consider the case of hyperbolic rational function, and more generally, functions postcritically finite on the Julia set without parabolics, that do not belong to the following list of exceptions: powers, Chebyshev polynomials and Latt‘es examples. In this case the quotient horospheric lamination is known to be minimal, while restricted to the union of nonisolated hyperbolic leaves (Glutsyuk, 2007). In the present paper we prove its unique ergodicity. To this end, we introduce the so-called transversely contracting flows and homeomorphisms (on abstract compact metrizable topological spaces), which include the vertical geodesic flows under consideration and the usual Anosov flows and diffeomorphisms. We prove a version of our unique ergodicity result for the transversely contracting flows and homeomorphisms. Particular cases for Anosov flows and diffeomorphisms are given by classical results by Bowen, Marcus, Furstenberg, Margulis, et al. We give a new and purely geometric proof, which seems to be simpler than the classical ones (which use either Markov partitions, K-property, or harmonic analysis).

Basic Properties of Metrizable Topological Spaces

Basic Properties of Metrizable Topological Spaces We continue Mizar formalization of general topology according to the book [11] by Engelking. In the article, we present the final theorem of Section 4.1. Namely, the paper includes the formalization of theorems on the correspondence between the cardinalities of the basis and of some open subcover, and a discreet (closed) subspaces, and the weight of that metrizable topological space. We also define Lindelöf spaces and state the above theorem in this special case. We also introduce the concept of separation among two subsets (see [12]).

Nonequality of dimensions for metric groups

An embeddability criterion for zero-dimensional metrizable topological spaces in zero-dimensional metrizable topological groups is given. A space which can be embedded as a closed subspace in a zero-dimensional metrizable group but is not strongly zero-dimensional is constructed; thereby, an example of a metrizable group with noncoinciding dimensions ind and dim is obtained. It is proved that one of Kulesza's zero-dimensional metrizable spaces cannot be embedded in a metrizable zero-dimensional group.

Semicontinuous representability of interval orders on a metrizable topological space

An interval order on an arbitrary set X is said to be representable by a pair (u, v) of real-valued functions on X if x y is equivalent to u(x) ≤ v(y) for all x, y ∈ X. We say that a topology τ on X is completely useful in the sense of an interval order (i.o. completely useful for the sake of brevity) if every upper semicontinuous interval order on the topological space (X, τ) is representable by a pair (u, v) of real-valued functions with v upper semicontinuous. We show that every i.o. completely useful topology on X is also completely useful (i.e., every upper semicontinuous total preorder on (X, τ) is representable by an upper semicontinuous utility function). Further we show that a metrizable topology τ on X is i.o. completely useful if and only if τ is separable.

Banach analytic structures on metrizable topological spaces and a cancellation problem

Let X be a topological space whose topology may be defined by a complete metric. Douady (case X compact) and Pestov (the general case) proved that X may be equipped with a structure of Banach analytic set. Here we prove that if X is not discrete, then this may be done in uncountably many ways. Let ( Y, d ) be a non-complete metric space and its completion. Here we prove that we may see as a closed analytic subset of a Banach space in such a way that with V a dense linear subspace of . Then we will prove (using bounded holomorphic functions) the following result. Let X be a connected open subset of an infinite-dimensional locally convex topological vector space V with the weak topology and A an open domain of such that has positive capacity. Then X × A is not biholomorphic to X.

Cocompactness and quasi-uniformizability of completely metrizable spaces

We show that a metrizable topological space X is completely metrizable if and only if it admits a quasi-uniformity U such that the topology induced by the conjugate quasi-uniformity U −1 on X is compact.