Gino Tironi

Weakly Fréchet–Urysohn and Pytkeev spaces

Abstract We study a new pointwise topological property, the weak Frechet–Urysohn property, introduced by Reznichenko. We also study the property introduced by Pytkeev in 1983. It is proved that sequentiality is strictly stronger than the Pytkeev property, which is strictly stronger than the wFU property, which is strictly stronger than countable tightness. However we prove that a countably tight compact Hausdorff space is Pytkeev. The above properties are used to detect some non-subsequential spaces.

The Lagrangian dynamics of thermal tracer particles in Navier-Stokes fluids

A basic issue for Navier-Stokes (NS) fluids is their characterization in terms of the so-called NS phase-space classical dynamical system, which provides a mathematical model for the description of the dynamics of infinitesimal (or ideal) tracer particles in these fluids. The goal of this paper is to analyze the properties of a particular subset of solutions of the NS dynamical system, denoted as thermal tracer particles (TTPs), whose states are determined uniquely by the NS fluid fields. Applications concerning both deterministic and stochastic NS fluids are pointed out. In particular, in both cases it is shown that in terms of the ensemble of TTPs a statistical description of NS fluids can be formulated. In the case of stochastic fluids this feature permits to uniquely establish the corresponding Langevin and Fokker-Planck dynamics. Finally, the relationship with the customary statistical treatment of hydrodynamic turbulence (HT) is analyzed and a solution to the closure problem for the statistical description of HT is proposed.

Pseudoradial spaces: Separable subsets, products and maps onto Tychonoff cubes

Abstract We work around our question of whether compact non-pseudoradial spaces have separable such subspaces. We obtain results about products of pseudoradial spaces and obtain more conditions which guarantee that each compact sequentially compact space is pseudoradial. We also discuss some questions of Sapirovskii which are also directed at separating the non-separable from the separable. We reinforce the need to focus on the space I ω 2 .

Compactifications and A-compactifications of frames. Proximal frames

Introduction The aim of this paper is to give two new descriptions of the ordered set (^J^(F), ^ ) of all (up to equivalence) regular compactifications of a completely regular frame F and to introduce and study the notion oi A-frame as a generalization of the notion of Alexandroff space (known also as zero-set space) (Alexandroff [1], Gordon[15]). A description of the ordered set of all (up to equivalence) Acompactifications of an A-frame by means of an ordered by inclusion set of some distributive lattices (called AP-sublattices) is obtained. It implies that any A-frame has a greatest A-compactification and leads to the descriptions of (J^Jf(-F), ^ ) . A new category U©5 isomorphic to the category ©ros^rm of proximal frames is introduced. A question for compactifications of frames analogous to the R. Chandlers question [8, p. 71] for compactifications of spaces is formulated and solved. Many results of [1], [3], [15], [23], [9], [10] and [11] are generalized. We first fix some notation. If C denotes a category, we write Xe \C\ if X is an object of C, and feC(X, Y) if / is a morphism of C with domain X and codomain Y. All lattices will be with top and bottom elements, denoted respectively by 1 and 0, and T)£at will stand for the category of distributive lattices and lattice homomorphisms. By an ordered set (M, ^ ) we mean a partially ordered set (i.e. ^ is a reflexive and transitive binary relation onM) for which ^ is also antisymmetric. If X is a set then we write expX for the set of all subsets of X. We denote by I the unit closed interval [0,1] with the natural topology, by Q the set of all rational numbers, by D> the set of all dyadic numbers in the interval (0,1), by N the set of all positive natural numbers and by 2 the simplest Boolean algebra.

First countable extensions of regular spaces

We investigate feebly compact extensions of first countable regular locally feebly compact spaces. Solutions of problems posed by R. M. Stephenson, Jr. and G. M. Reed are given.

d-4 – Pseudoradial Spaces

Publisher Summary The class of pseudo radial spaces, with the name of folgenbestimmte Raume, was introduced by H. Herrlich in 1967; they have also been called chain-net spaces. They are defined using the convergence of transfinite sequences. A (transfinite) sequence is a map whose domain is an ordinal. In these cases, only regular initial ordinals need to be considered, that is, regular cardinal numbers. Some subclasses of the class of pseudo radial spaces were introduced successively because they were useful for solving natural problems that arise for pseudo radial spaces. The class of almost radial spaces came first. Herrlich proved that the pseudo radial spaces are quotients of linearly ordered topological spaces and are quotients of spaces in which every point has a well ordered local base of neighborhoods. All Frechet–Urysohn spaces are sequential and radial. All radial and all sequential spaces are almost radial, and all almost radial spaces are pseudo radial. One of the first problems considered in connection with the notion of pseudo radialness was to find Zermelo–Fraenkel set theory with the axiom of choice (ZFC) examples of non-sequential pseudo radial spaces with countable tightness. It is evident from Baloghs theorem that compact spaces of countable tightness are sequential.

Compactifications, A-compactifications and proximities

A functor S from the category RegσFrm of regular σ-frames to the category DLat of distributive lattices is defined. The notion of AP-sublattice of S(α), for α ∈ ¦RegσFrm¦, is introduced and it is shown that, for every Alexandroff space (X, α), the ordered set of all (up to equivalence) A-compactifications of (X, α) is isomorphic to the set of all AP-sublattices of S(α) ordered by inclusion. This gives, in particular, a description of the ordered set of z-compactifications of a T3 1/2-space. Further, for any Tychonoff space X, the notion of a P-sublattice of S (CozX) is defined and it is proved that the ordered set of all (up to equivalence) T2-compactifications of X is isomorphic to the set of all P-sublattices of S(CozX) ordered by inclusion. We construct proximities by means of AP- and P-sublattices. Moreover, using these notions, we introduce two concrete categories PHA and PASF which are respectively isomorphic and dual to the category Prox of proximity spaces.

The Mathematical Modelling of Thermal Tracer Particles in Navier-Stokes Fluids

In this paper we report the discovery of a subset of socalled ideal tracer particles (ITP (4)) belonging to NavierStokes (NS) fluids which are denoted as thermal tracer particles (TTP). Their states are found to be uniquely dependent on the local state of the fluid. This means that a suitable statistical ensemble of TTPs should reproduce exactly the dynamics of the fluid. In other words, it should be possible to determine the fluid fields characterizing the fluid state by means of suitable statistical averages on the ensemble of TTPs so that they satisfy identically the required set of fluid equations. The result applies to NS fluids described as mesoscopic, i.e., continuous fluids, which can be either viscous or inviscid, compressible or incompressible, thermal or isothermal, isentropic or nonisentropic. We shall assume that the state of these fluids is represented by an ensemble of observables {Z(r, t)} ≡ {Zi(r, t), i = 1, .., n} (with n an integer ≥ 1), i.e., fluid fields, which can be unambiguously prescribed as continuous and suitably smooth functions, respectively, in Ω × I and in the open set Ω × I (in the following to be identified with a bounded subset of R). We intend to show that, as a remarkable consequence, the phasespace dynamical system which advances in time (the states of) these particles can be uniquely prescribed in such a way to determines self-consistently the time evolution of the complete set of fluid equations characterizing the fluid. This implies that TTPs must reproduce exactly the dynamics of the fluid. In other words, by means of an appropriate statistical averages on the ensemble of the TTPs, it is possible to determine the time-evolution of the fluid state, in such a way that it satisfies identically the required set of fluid equations. In this paper we point out that the existence of TTPs can be reached by means of a Gedanken experiment on the NS fluid, i.e., by inspecting the properties of the underlying clsssical dynamical system which advances in time the state of a NS fluid.

Frames and Grids

M. Barr and M.-C. Pedicchio introduced the category Grids of grids in order to show that the opposite of the category Top of topological spaces is a quasivariety. J. Adámek and M.-C. Pedicchio proved that there exists a duality D between the category TopSys of topological systems (defined by S. Vickers) and the category Grids. In both papers a description of the full subcategory D(Top) of the category Grids is given. In this paper we describe internally all grids isomorphic to the objects of the full coreflective subcategory D(Loc) of the category Grids, i.e. we characterize internally all grids of the form D(C), where C is a localic topological system (here Loc is the category of locales regarded as a full subcategory of TopSys). Since, obviously, the category Frm of frames is equivalent to D(Loc), we can say that in this paper those grids which could be called frames are characterized internally. An internal characterization of all grids which correspond (in the above sense) to the frames having T1 spectra and a generalization of the well-known fact that the spectrum of a locale is a sober space are obtained as well.