Related papers: On the Mackey problem for free locally convex spac…

We give a criterion for a set of $n$ hyperbolic isometries of a $\mathrm{CAT}(0)$ metric space $X$ to generate a free group on $n$ generators. This extends a result by Alperin, Farb and Noskov who proved this for 2 generators under the…

A topological space $X$ is $\kappa$-Fr\'{e}chet--Urysohn if for every open subset $U$ of $X$ and every $x\in \overline{U}$ there exists a sequence in $ U$ converging to $x$. We prove that every $\kappa$-Fr\'{e}chet--Urysohn Tychonoff space…

To study induced representation of some class of groups, Mackey's theory is very useful. In this paper, we consider some generalization of Mackey's theory for locally profinite groups. In particular, we give conditions on groups under which…

There is a hierarchy of structure conditions for convex sets. In this paper we study a recently defined [3, 8, 9] condition called locally nonconical convexity (abbreviated LNC). Is is easy to show that every strictly convex set is LNC, as…

A coarse space $X$, endowed with a linear order compatible with the coarse structure of $X$, is called linearly ordered. We prove that every linearly ordered coarse space $X$ is locally convex and the asymptotic dimension of $X$ is either…

We provide necessary and/or sufficient conditions on vector spaces $V$ of real sequences to be a Fr\'{e}chet space such that each coordinate map is continuous, that is, to be a locally convex FK space. In particular, we show that if…

A topological space $X$ is called submaximal if every dense subset of $X$ is open. In this paper, we show that if $\beta X$, the Stone-\v{C}ech compactification of $X$, is a submaximal space, then $X$ is a compact space and hence $\beta…

In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of…

We show that every limit point of a Zariski dense discrete subgroup $\Gamma$ of the isometry group of a symmetric space of noncompact type is conical if and only if $\Gamma$ is convex cocompact.

We prove that: I. If $L$ is a $T_1$ space, $|L|>1$ and $d(L) \leq \kappa \geq \omega$, then there is a submaximal dense subspace $X$ of $L^{2^\kappa}$ such that $|X|=\Delta(X)=\kappa$; II. If $\frak{c}\leq\kappa=\kappa^\omega<\lambda$ and…

We study topological versions of an independent set in an abelian group and a linearly independent set in a vector space, a {\em topologically independent set} in a topological group and a {\em topologically linearly independent set} in a…

We consider classes T of topological spaces (referred to as T-spaces) that are stable under continuous images and frequently under arbitrary products. A local T-space has for each point a neighborhood base consisting of subsets that are…

We extend the well-known Gelfand-Phillips property for Banach spaces to locally convex spaces, defining a locally convex space $E$ to be Gelfand-Phillips if every limited set in $E$ is precompact in the topology on $E$ defined by barrels.…

Let $X$ be an Abelian group of the form $X=\mathbb{R}^m\times K\times D$, where $m\geq 0$, $K$ is a compact totally disconnected group of the special form, $D$ is a discrete group. Let $\xi_i, i=1,2,...,n,n\geq 2,$ be independent random…

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…

It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here we prove that a discrete version of this property (called local to global rigidity) holds for a…

We characterize topological (and uniform) spaces whose free (locally convex) topological vector spaces have a local $\mathfrak G$-base. A topological space $X$ has a local $\mathfrak G$-base if every point $x$ of $X$ has a neighborhood base…

Motivated by the theory of locally definable groups, we study the theory of $K$-vector spaces with a predicate for the union $X$ of an infinite family of independent subspaces. We show that if $K$ is infinite then the theory is complete and…

In this paper, we introduce Mackey functors for a t.d.l.c. group and define the cohomological dimension of this group over the Mackey category. We then compare this dimension to the rational discrete cohomological dimension defined by…

We present a constructive proof of Tychonoff's fixed point theorem in a locally convex space for sequentially locally non-constant functions, As a corollary to this theorem we also present Schauder's fixed point theorem in a Banach space…