In this paper, we introduce and study two new classes of subrings of C(X): norm-closed subrings and norm-reflecting subrings. A subring ℝ ⊆ S ⊆ C ( X ) is said to be norm-closed if for every f ∈ S , the function |f| ∈ S; it is norm-reflecting if |f| ∈ S implies f ∈ S. These concepts are inspired by the lattice structure of C(X), particularly the operation of taking absolute values. We provide characterizations of norm-closed (norm-reflecting) subrings. Several examples and counterexamples are presented to illustrate the distinctions and connections between these classes.

We consider various strengthenings of the notion of topological transitivity in non-autonomous discrete dynamical systems. We give many equivalent conditions foreach of these notions and present the implications among them. We also considerrearrangements of non-autonomous dynamical systems.

Let X be an infinite dimensional separable Banach space, T : X → X be a hypercyclic operator, and x ∈ X be a (frequently) hypercyclic vector of T. We show that if the terms from the T-orbit of x converge to a vector y sufficiently fast, then y is also a hypercyclic vector of T. As a corollary, we deduce that if T is a frequently hypercyclic operator with spectral radius r(T) = 1, then lim_{n\to \infty} ∥Tnx∥^{1/n} = 1 for every frequently hypercyclic vector x of T. Some related observations are also made.

We prove that each semitopological semigroup has a reflection in the class of abelian cancellative semitopological semigroups. Then we use this reflection to prove that the group of characters of a locally compact pseudocompact topological semigroup with open shifts is discrete.

The topic of fixed points in digital metric spaces continues to draw publications with assertions that are incorrect, incorrectly proven, trivial, or incoherently stated. We continue the work of our earlier papers that discuss publications with bad assertions concerning fixed points of selffunctions on digital images.

The purpose of this paper is to present the construction of the completion of a set equipped with a modular metric that we call the modular completion. We show for instance that any set equipped with a modular metric has a modular completion. Moreover, it turns out that the modular completion is unique up to isometry.

Given a metric continuum X, the hyperspace of meager subcontinua of X, denoted by M(X), is the collection of all subcontinua of X with empty interior. Answering questions by Norberto Ordo˜nez, in this paper we construct two dendroids showing that there is no implication between the conditions: (a) X is contractible and (b) M(X) is contractible. We also prove that if X is an arc-smooth continuum, then M(X) is contractible.

In this paper we introduce the almost strongly star-Menger property and we provide some results and relationships with another known properties in literature. Furthermore, we characterize the almost strongly star-Menger and the almost star-Menger properties in hyperspaces endowed with the hit-and-miss topology $\tau_\Delta$, by using two new selection principles $\mathbf{S}_{\textsf{fin}}^{\star}(\Pi_{\Delta}(\Lambda), \mathscr{B})$ and $\mathbf{SS}_{\textsf{fin}}^{\star}(\Pi_{\Delta}(\Lambda), \mathscr{B})$, for an appropriate family $\mathscr{B}$.

We study spaces X for which the space Homp(X) of automorphisms with the topology of point-wise convergence is a topological group. We identify large classes of spaces X for which Homp(X) is or is not a topological group.

In this paper, we point out mathematical errors and flaws in the proofs of certain recent results obtained by some authors in relational metric spaces. We also provide corrected versions of these proofs, thereby improving the accuracy and reliability of the corresponding results.