Related papers: Sequences suffice for pointfree uniform completion…
We use a generic notion of flatness in the enriched context to define various completions of metric spaces -- enrichments over [0,\infty] -- and preorders -- enrichments over 2. We characterize the weights of colimits commuting in…
In this short note, we obtain partial quasi-metric versions of Kannan's fixed point theorem for self-mappings. Moreover, we use these fixed points results to characterize a certain type of completeness in partial quasi-metric spaces. We…
The famous Banach Contraction Principle holds in complete metric spaces, but completeness is not a necessary condition -- there are incomplete metric spaces on which every contraction has a fixed point. The aim of this paper is to present…
In this note a criterion for Cauchy sequences is proved which refines the one presented in `Cauchy sequences in b-metric spaces', Topology Appl. 373 (2025) 109477.
We consider a fixed basis of a finitely generated free chain complex as a finite topological space and we present a sufficient condition for the singular homology of this space to be isomorphic with the homology of the chain complex.
In these expository notes, intended for students without background in point-set topology, we develop the basic theory of the Stone-Cech compactification without reference to open sets, closed sets, filters, or nets. In particular, this…
A topological space is called a submetrizable if it can be mapped onto a metrizable topological space by a continuous one-to-one map. In this paper we answer two questions concerning sequence-covering maps on submetrizable spaces.
The aim of this paper is to investigate the equivalence conditions for uniform perfectness of quasi-metric spaces. We also obtain the invariant property of uniform perfectness under quasim\"obius maps in quasi-metric spaces. In the end, two…
In this oaper, we prove some fixed point theorems in metric vector spaces, in which the continuity is not required for the considered mappings to satisfy. We provide some concrete examples to demonstrate these theorems. We also give some…
In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to $\R^n$. A natural question arises, whether these approaches are equivalent…
Many concrete problems are formulated in terms of a finite set of points in $R^n$ which, via the ambient Euclidean metric, becomes a finite metric space. To obtain information from such a space, it is often useful to associate a graph to…
This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of…
We consider the Cauchy problem for the wave equation in the whole space, R^n, with initial data which are distributions supported on finite sets. The main result is a precise description of the geometry of the sets of stationary points of…
The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss…
This paper provides an Open Mapping Theorem for topological modules over rings that have a zero sequence consisting of units. As an application it is shown that there is a unique complete and metrisable topology on finitely generated…
A bounded subset of a normed linear space is said to be (diametrically) complete if it cannot be enlarged without increasing the diameter. A complete super set of a bounded set $K$ having the same diameter as $K$ is called a completion of…
Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for…
A metric space $\mathbf{X}$ is called densely complete if there exists a dense set $D$ in $\mathbf{X}$ such that every Cauchy sequence of points of $D $ converges in $\mathbf{X}$. One of the main aims of this work is to prove that the…
We study the completeness and ultracompleteness numbers of a convergence space. In the case of a completely regular topological space, the completeness number is countable if and only if the space is $\v{C}$ech-complete, and the…
The language of finite games is used to rephrase Pelant's proof of his result: The separable modification of the complete metric space $C([0,\omega_1])$ is not complete.