Contramodules over pro-perfect topological rings

For four wide classes of topological rings R \mathfrak{R} , we show that all flat left R \mathfrak{R} -contramodules have projective covers if and only if all flat left R \mathfrak{R} -contramodules are projective if and only if all left R \mathfrak{R} -contramodules have projective covers if and only if all descending chains of cyclic discrete right R \mathfrak{R} -modules terminate if and only if all the discrete quotient rings of R \mathfrak{R} are left perfect. Three classes of topological rings for which this holds are the complete, separated topological associative rings with a base of neighborhoods of zero formed by open two-sided ideals such that either the ring is commutative, or it has a countable base of neighborhoods of zero, or it has only a finite number of semisimple discrete quotient rings. The fourth class consists of all the topological rings with a base of neighborhoods of zero formed by open right ideals which have a closed two-sided ideal with certain properties such that the quotient ring is a topological product of rings from the previous three classes. The key technique on which the proofs are based is the contramodule Nakayama lemma for topologically T-nilpotent ideals.

Finitary approximations of coarse structures

A coarse structure $ \mathcal{E}$ on a set $X$ is called finitary if, for each entourage $E\in \mathcal{E}$, there exists a natural number $n$ such that $ E[x]< n $ for each $x\in X$. By a finitary approximation of a coarse structure $ \mathcal{E}^\prime$, we mean any finitary coarse structure $ \mathcal{E}$ such that $ \mathcal{E}\subseteq \mathcal{E}^\prime$.If $\mathcal{E}^\prime$ has a countable base and $E[x]$ is finite for each $x\in X$ then $ \mathcal{E}^\prime$has a cellular finitary approximation $ \mathcal{E}$ such that the relations of linkness on subsets of $( X,\mathcal{E}^\prime)$ and $( X, \mathcal{E})$ coincide.This answers Question 6 from [8]: the class of cellular coarse spaces is not stable under linkness. We define and apply the strongest finitary approximation of a coarse structure.

FLAT RING EPIMORPHISMS OF COUNTABLE TYPE

Let R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology $\mathbb{G}$ of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at $\mathbb{G}$ fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology $\mathbb{G}$, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all $\mathbb{G}$-separated $\mathbb{G}$-complete left R-modules.

Weakly Almost Periodic Points and Some Chaotic Properties of Dynamical Systems

Let X be a compact metric space and f : X → X be a continuous map. In this paper, ergodic chaos and strongly ergodic chaos are introduced, and it is proven that f is strongly ergodically chaotic if f is transitive but not minimal and has a full measure center. In addition, some sufficient conditions for f to be Ruelle–Takens chaotic are presented. For instance, we prove that f is Ruelle–Takens chaotic if f is transitive and there exists a countable base [Formula: see text] of X such that for each i > 0, the meeting time set N(Ui, Ui) for Ui with respect to itself has lower density larger than [Formula: see text].

On Convergence with respect to an Ideal and a Family of Matrices

P. Das et al. recently introduced and studied the notions of strong AI-summability with respect to an Orlicz function F and AI-statistical convergence, where A is a nonnegative regular matrix and I is an ideal on the set of natural numbers. In this paper, we will generalise these notions by replacing A with a family of matrices and F with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' sup-limsup-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal I has a countable base), continuing some of the author's previous work.

Regular bases at non-isolated points and metrization theorems

In this paper, we define the spaces with a regular base at non-isolated points and discuss some metrization theorems. We firstly show that a space X is a metrizable space, if and only if X is a regular space with a σ-locally finite base at non-isolated points, if and only if X is a perfect space with a regular base at non-isolated points, if and only if X is a β-space with a regular base at non-isolated points. In addition, we also discuss the relations between the spaces with a regular base at non-isolated points and some generalized metrizable spaces. Finally, we give an affirmative answer for a question posed by F. C. Lin and S. Lin in [7], which also shows that a space with a regular base at non-isolated points has a point-countable base.

A NOTE ON STAR COMPACT SPACES WITH POINT-COUNTABLE BASE

In this note we give an example of a Hausdorff, star compact space with point-countable base which is not metrizable.

A Note on Topological Properties of Non-Hausdorff Manifolds

The notion of compatible apparition points is introduced for non-Hausdorff manifolds, and properties of these points are studied. It is well known that the Hausdorff property is independent of the other conditions given in the standard definition of a topological manifold. In much of literature, a topological manifold of dimension is a Hausdorff topological space which has a countable base of open sets and is locally Euclidean of dimension . We begin with the definition of a non-Hausdorff topological manifold.

Strong and Uniform Continuity – the Uniform Space Case

It is proved, within the constructive theory of apartness spaces, that a strongly continuous mapping from a totally bounded uniform space with a countable base of entourages to a uniform space is uniformly continuous. This lifts a result of Ishihara and Schuster from metric to uniform apartness spaces. The paper is part of a systematic development of computable topology using apartness as the fundamental notion.

Diffraction of Weighted Lattice Subsets

A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice Γ inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniformlattice Dirac comb, and its diffraction measure is periodic, with the dual lattice Γ*as lattice of periods. This statement remains true in the setting of a locally compact Abelian group whose topology has a countable base.