Related papers: $\pi$-metrizable spaces and strongly $\pi$-metriza…

A consistent path system in a graph $G$ is an collection of paths, with exactly one path between any two vertices in $G$. A path system is said to be consistent if it is intersection-closed. We say that $G$ is strictly metrizable if every…

We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure $\mathfrak{p}$ and a volume form $\sigma$ on an oriented surface $M$ equip the total space…

A space has $\sigma$-compact tightness if the closures of $\sigma$-compact subsets determines the topology. We consider a dense set variant that we call densely k-separable. We consider the question of whether every densely k-separable…

A metric space is said to be strongly rigid if no positive distance is taken twice by the metric. In 1972, Janos proved that a separable metrizable space has a strongly rigid metric if and only if it is zero-dimensional. In this paper, we…

Let $\mathcal{X}$ be a smooth Artin stack with properly stable good moduli space $\pi\colon\mathcal{X} \to X$. The purpose of this paper is to prove that a simple geometric criterion can often characterize when the moduli space $X$ is…

A topological space $X$ is defined to have a neighborhood $P$-base at any $x\in X$ from some poset $P$ if there exists a neighborhood base $(U_p[x])_{p\in P}$ at $x$ such that $U_p[x]\subseteq U_{p'}[x]$ for all $p\geq p'$ in $P$. We prove…

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in G$…

If $X$ is a (topological) space, the $n$th finite subset space of $X$, denoted by $X(n)$, consists of $n$-point subsets of $X$ (i.e., nonempty subsets of cardinality at most $n$) with the quotient topology induced by the unordering map…

On a smooth connected manifold, we consider all possible locally elliptic and locally bounded measurable coefficient Riemannian metrics called rough Riemannian metrics. We equip this set with an extended metric which is connected if and…

In this article we studied the relationship between metric spaces and multiplicative metric spaces. Also, we pointed out some fixed and common fixed point results under some contractive conditions in multiplicative metric spaces can be…

It is well-known that a metric space $(X, d)$ is complete iff the set $X$ is closed in every metric superspace of $(X, d)$. For a given pseudometric space $(Y, \rho)$, we describe the maximal class $\mathbf{CEC}(Y, \rho)$ of superspaces of…

We study metric spaces that admit a conical bicombing and thus obey a weak form of non-positive curvature. Prime examples of such spaces are injective metric spaces. In this article we give a complete characterization of complete metric…

We characterize conformally removable sets in the plane with the aid of the recent developments in the theory of metric surfaces. We prove that a compact set in the plane is $S$-removable if and only if there exists a quasiconformal map…

Topologies $\tau, \sigma$ on a set $X$ are called $\pi$-compatible if $\tau$ is a $\pi$-network for $\sigma$, and vice versa. If topologies $\tau, \sigma$ on a set $X$ are $\pi$-compatible then the families of nowhere dense sets (resp.…

In this paper, we study isometries of $p$-Wasserstein spaces. In our first result, for every complete and separable metric space $X$ and for every $p\geq1$, we construct a metric space $Y$ such that $X$ embeds isometrically into $Y$, and…

In spacetime physics, we frequently need to consider a set of all spaces (`universes') as a whole. In particular, the concept of `closeness' between spaces is essential. However, there has been no established mathematical theory so far…

We introduce a new class of $\varkappa$-metrizable spaces, namely countably $\varkappa$-metrizable spaces. We show that the class of all $\varkappa$-metrizable spaces is a proper subclass of counably $\varkappa$-metrizable spaces. On the…

For a metrizable space $X$ of density $\kappa$, let $PM(X)$ be the space of continuous bounded pseudometrics on $X$ endowed with the uniform convergence topology. In this paper, its topology shall be classified as follows: (i) If $X$ is…

Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial…

A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first…