Related papers: Metrizable DH-spaces with a dense complete subset

This note tries to give an answer to the following question: Is there a sufficiently rich class of metric vector spaces such that sufficiently large spaces of continuous linear maps between them are metrizable?

A space $X$ is submaximal if any dense subset of $X$ is open. In this paper, we prove that every submaximal topological gyrogroup of non-measurable cardinality is strongly $\sigma$-discrete. Moreover, we prove that every submaximal strongly…

For a space $X$, let $(CL(X), \tau_V)$, $(CL(X), \tau_{locfin})$ and $(CL(X), \tau_F)$ be the set $CL(X)$ of all nonempty closed subsets of $X$ which are endowed with Vietoris topology, locally finite topology and Fell topology…

We prove several results of the following type: given finite dimensional normed space V there exists another space X with log (dim X) = O(log (dim V)) and such that every subspace (or quotient) of X, whose dimension is not "too small,"…

A topological space $X$ is selectively highly divergent (SHD) if for every sequence of non-empty open subsets $\{U_n: n\in \omega \}$ of $X$, we can pick a point $x_n\in U_n$, for every $n<\omega$, such that the sequence $\{x_n:…

We present a necessary and sufficient condition for the strict positive definiteness of a real, continuous, isotropic and positive definite kernel on a two-point compact homogeneous space. The characterization adds to others previously…

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…

Given a Banach space we consider the $\sigma$-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the…

A 7-dimensional area-minimizing embedded hypersurface $M$ will in general have a discrete singular set. The same is true if $M$ is stable, or has bounded index, provided $H^6(sing M) = 0$. We show that if $M_i$ are a sequence of such…

In this paper, the controllability and observability of linear multi-agent systems over matrix-weighted signed networks are analyzed. Firstly, the definition of equitable partition of matrix-weighted signed multi-agent system is given, and…

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a…

In this paper, a necessary and sufficient condition for the controllability of networked systems with heterogeneous dynamics is established where the nodes are higher dimensional linear time invariant systems and the network topology is…

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and…

The study of controlled hybrid systems requires practical tools for approximation and comparison of system behaviors. Existing approaches to these problems impose undue restrictions on the system's continuous and discrete dynamics.…

Let $G$ be a semisimple real Lie group with finite center and $H$ a connected closed subgroup. We establish a geometric criterion which detects whether the representation of $G$ in $L^2(G/H)$ is tempered.

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 $\sigma$-locally…

We introduce the property of countable separation for a locally convex Hausdorff space $X$ and relate it to the existence of a metrizable coarser topology. Building on this, we demonstrate how the separability of $X$ is equivalent to the…

A digraph is connected-homogeneous if every isomorphism between two finite connected induced subdigraphs extends to an automorphism of the whole digraph. In this paper, we completely classify the countable connected-homogeneous digraphs.

We define a simple criterion for a homogeneous, complete metric structure $X$ that implies that the automorphism group $\mbox{Aut}(X)$ satisfies all the main consequences of the existence of ample generics: it has the small index property,…

Building on work of Terada, we prove that h-homogeneity is productive in the class of zero-dimensional spaces. Then, by generalizing a result of Motorov, we show that for every non-empty zero-dimensional space $X$ there exists a non-empty…