On locally LC-spaces

Asymptotic dimension and small subsets in locally compact topological groups

We prove that for a coarse space $X$ the ideal $S(X)$ of small subsets of $X$ coincides with the ideal $D_<(X)$ of subsets $A\subset X$ of asymptotic dimension $asdim(A)<asdim(X)$ provided that $X$ is coarsely equivalent to an Euclidean space $R^n$. Also we prove that for a locally compact Abelian group $X$, the equality $S(X)=D_<(X)$ holds if and only if the group $X$ is compactly generated.

$P$-Paracompact and $P$-Metrizable Spaces

Let $P$ be a directed set and $X$ a space. A collection $\mathcal{C}$ of subsets of $X$ is \emph{$P$-locally finite} if $\mathcal{C}=\bigcup \{ \mathcal{C}_p : p \in P\}$ where (i) if $p \le p'$ then $\mathcal{C}_p \subseteq \mathcal{C}_{p'}$ and (ii) each $\mathcal{C}_p$ is locally finite. Then $X$ is \emph{$P$-paracompact} if every open cover has a $P$-locally finite open refinement. Further, $X$ is \emph{$P$-metrizable} if it has a $(P \times \mathbb{N})$-locally finite base. This work provides the first detailed study of $P$-paracompact and $P$-metrizable spaces, particularly in the case w…

Universal nowhere dense subsets of locally compact manifolds

In each manifold $M$ modeled on a finite or infinite dimensional cube $[0,1]^n$ we construct a closed nowhere dense subset $S\subset M$ (called a spongy set) which is a universal nowhere dense set in $M$ in the sense that for each nowhere dense subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h(A)\subset S$. The key tool in the construction of spongy sets is a theorem on topological equivalence of certain decompositions of manifolds. A special case of this theorem says that two vanishing cellular strongly shrinkable decompositions $\mathcal A,\mathcal B$ of a Hilbert cube man…

A remark on $β$-locally closed sets

The aim of this note is to show that every subset of a given topological space is the intersection of a preopen and a preclosed set, therefore $β$-locally closed, and that every topological space is $β$-submaximal.

Irreducible Quotient Maps From Locally Compact Separable Metric Spaces

Let X be a Hausdorff quotient of a standard space (that is of a locally compact separable metric space). It is shown that the following are equivalent: (i) X is the image of an irreducible quotient map from a standard space; (ii) X has a sequentially dense subset satisfying two technical conditions involving double sequences; (iii) whenever q : Y\to X is a quotient map from a standard space Y , the restriction q_{\st}|V is an irreducible quotient map from V onto X (where q_{\st} : Y_{\st}\to X is the pure quotient derived from q, and V is the closure of the set of singleton fibres of Y_{\st}).…

Haar-open sets: a right way of generalizing the Steinhaus sum theorem to non-locally compact groups

Let $X$ be the countable product of Abelian locally compact Polish groups and $A,B\subset X$ be two Borel sets, which are not Haar-null in $X$. We prove that the sum-set $A+B:=\{a+b:a\in A,\;\;b\in B\}$ is Haar-open in the sense that for any non-empty compact subset $K\subset X$ and point $p\in K$ there exists a point $x\in X$ such that the set $K\cap(A+B+x)$ is a neighborhood of $p$ in $K$. This is a generalization of the classical Steinhaus Theorem (1920) to non-locally compact groups. We do not know if this generalization holds for Banach spaces.

The compact strong Z-set property in a hyperspace of finite subsets

Let X be a non-degenerate, connected, locally path-connected metrizable space and Fin(X) be the hyperspace consisting of non-empty finite subsets in X endowed with the Vietoris topology. In this paper, we show that every compact set in Fin(X) is a strong Z-set.

Locally minimal topological groups

A Hausdorff topological group $(G,τ)$ is called locally minimal if there exists a neighborhood $U$ of 0 in $τ$ such that $U$ fails to be a neighborhood of zero in any Hausdorff group topology on $G$ which is strictly coarser than $τ.$ Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric…

Locally solid topological lattice-ordered groups

Locally solid Riesz spaces have been widely investigated in the past several decades; but locally solid topological lattice-ordered groups seem to be largely unexplored. The paper is an attempt to initiate a relatively systematic study of locally solid topological lattice-ordered groups. We give both Robert-Namioka-type characterization and Fremlin-type characterization of locally solid topological lattice-ordered groups. In particular, we show that a group topology on a lattice-ordered group is locally solid if and only if it is generated by a family of translation-invariant lattice pseudomet…

The Gelfand-Phillips property for locally convex spaces

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. Several characterizations of Gelfand-Phillips spaces are given. The problem of preservation of the Gelfand-Phillips property by standard operations over locally convex spaces is considered. Also we explore the Gelfand-Phillips property in spaces $C(X)$ of continuous functions on a Tychonoff space $X$. If $τ$ and $\mathcal T$ are two locally convex topologies …

An example of a non-Borel locally-connected finite-dimensional topological group

Answering a question posed by S.Maillot in MathOverFlow, for every $n\in\mathbb N$ we construct a locally connected subgroup $G\subset\mathbb R^{n+1}$ of dimension $dim(G)=n$, which is not locally compact.

Locally Contractive Maps on Perfect Polish Ultrametric Spaces

In this paper we present a result concerning locally contractive maps defined on subsets of perfect Polish ultrametric spaces (i.e. separable complete ultrametric spaces). Specifically, we show that a perfect compact ultrametric space cannot be contained in its locally contractive image, a corollary relating this result to minimal dynamical systems, and pose a conjecture for the general Polish ultrametric case.

Locally convex spaces with the strong Gelfand-Phillips property

We introduce the strong Gelfand-Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand-Phillips property among locally convex spaces admitting a stronger Banach space topology. If $C_{\mathcal T}(X)$ is a space of continuous functions on a Tychonoff space $X$, endowed with a locally convex topology $\mathcal T$ between the pointwise topology and the compact-open topology, then: (a) the space $C_{\mathcal T}(X)$ has the strong Gelfand-Phillips property iff $X$ contains a compact countable subspace $K\subseteq X$ of fin…

Closed locally path-connected subspaces of finite-dimensional groups are locally compact

We prove that each closed locally continuum- connected subspace of a finite dimensional topological group is locally compact. This allows us to construct many 1-dimensional metrizable separable spaces that are not homeomorphic to closed subsets of finite-dimensional topological groups, which answers in negative a question of D.Shakhmatov. Another corollary is a characterization of Lie groups as finite-dimensional locally continuum-connected topological groups. For locally path connected topological groups this characterization was proved by Gleason and Palais in 1957.

局部紧群的纤维化与陪集空间

Fibrations and coset spaces for locally compact groups

本文证明当闭子群 $L$ 为局部紧 pro-Lie 群时，陪集空间映射 $G/K\to G/L$ 是纤维化，并由此推导出 Skljarenko、Madison 和 Mostert 的两个经典结论。

Tameness and Rosenthal type locally convex spaces

Motivated by Rosenthal's famous $l^1$-dichotomy in Banach spaces, Haydon's theorem, and additionally by recent works on tame dynamical systems, we introduce the class of tame locally convex spaces. This is a natural locally convex analogue of Rosenthal Banach spaces (for which any bounded sequence contains a weak Cauchy subsequence). Our approach is based on a bornology of tame subsets which in turn is closely related to eventual fragmentability. This leads, among others, to the following results: $\bullet$ extending Haydon's characterization of Rosenthal Banach spaces, by showing that a lcs $…

Some Generalizations of Fedorchuk Duality Theorem -- II

As it was shown in the first part of this paper, there exists a duality between the category DSkeLC (introduced there) and the category SkeLC of locally compact Hausdorff spaces and continuous skeletal maps. We describe here the subcategories of the category DSkeLC which are dually equivalent to the following eight categories: all of them have as objects the locally compact Hausdorff spaces and their morphisms are, respectively, the injective (respectively, surjective) continuous skeletal maps, the injective (resp., surjective) open maps, the injective (resp., surjective) skeletal perfect maps…

Locally precompact groups: (Local) realcompactness and connectedness

A theorem of A. Weil asserts that a topological group embeds as a (dense) subgroup of a locally compact group if and only if it contains a non-empty precompact open set; such groups are called locally precompact. Within the class of locally precompact groups, the authors classify those groups with the following topological properties: Dieudonné completeness; local realcompactness; realcompactness; hereditary realcompactness; connectedness; local connectedness; zero-dimensionality. They also prove that an abelian locally precompact group occurs as the quasi-component of a topological group if a…

The $κ$-Fréchet--Urysohn property for locally convex spaces

A topological space $X$ is $κ$-Fréchet--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 $κ$-Fréchet--Urysohn Tychonoff space $X$ is Ascoli. We apply this statement and some of known results to characterize the $κ$-Fréchet--Urysohn property in various important classes of locally convex spaces. In particular, answering a question posed in [7] we obtain that $C_p(X)$ is Ascoli iff $X$ has the property $(κ)$.

Mazurkiewicz manifolds and homogeneity

It is proved that no region of a homogeneous locally compact, locally connected metric space can be cut by an $F_σ$-subset of a "smaller" dimension. The result applies to different finite or infinite topological dimensions of metrizable spaces.

The group of isometries of a locally compact metric space with one end

In this note we study the dynamics of the natural evaluation action of the group of isometries $G$ of a locally compact metric space $(X,d)$ with one end. Using the notion of pseudo-components introduced by S. Gao and A. S. Kechris we show that $X$ has only finitely many pseudo-components of which exactly one is not compact and $G$ acts properly on. The complement of the non-compact component is a compact subset of $X$ and $G$ may fail to act properly on it.

Locally quasi-convex topologies on the group of the integers

\noindent The most natural group topology on $\Z$ is the discrete one. There are other well-known group topologies on $\Z$, like the $p$-adic, defined for any prime number $p$. It is also an important group topology the weak topology with respect to the group of homomorphisms from $\Z$ to the unit circle of the complex plane; that is, the one defined by the characters and which is known as "the Bohr topology" on $\Z$. \noindent In \cite{tesislorenzo}, it is proved that taking as a neighbourhood basis at 0 the subsets $\{W_n\mid n\in\N\}$, defined by $W_n:=\{k\in\Z\mid\forall x\in S$, $ k\cdot …

局部紧强拓扑gyrogroup中合适集的存在性

The existence of suitable sets in locally compact strongly topological gyrogroups

本文证明了每个局部紧强拓扑gyro群都存在一个合适集，解决了Lin等人提出的问题。

Building suitable sets for locally compact groups by means of continuous selections

If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S \cup {1} is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory.

Topological structure of non-separable sigma-locally compact convex sets

For an infinite cardinal $κ$ let $\ell_2(κ)$ be the linear hull of the standard othonormal base of the Hilbert space $\ell_2(κ)$ of density $κ$. We prove that a non-separable convex subset $X$ of density $κ$ in a locally convex linear metric space if homeomorphic to the space (i) $\ell_2^f(κ)$ if and only if $X$ can be written as countable union of finite-dimensional locally compact subspaces, (ii) $[0,1]^ω\times \ell_2^f(κ)$ if and only if $X$ contains a topological copy of the Hilbert cube and $X$ can be written as a countable union of locally compact subspaces.

$\mathfrak G$-bases in free (locally convex) topological vector spaces

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 $(U_α)_{α\inω^ω}$ such that $U_β\subset U_α$ for all $α\leβ$ in $ω^ω$. To construct $\mathfrak G$-bases in free topological vector spaces, we exploit a new description of the topology of a free topological vector space over a topological (or more generally, uniform) space.

Points fixes des applications compactes dans les espaces ULC

A topological space is locally equiconnected if there exists a neighborhood $U$ of the diagonal in $X\times X$ and a continuous map $λ:U\times[0,1]\to X$ such that $λ(x,y,0)=x$, $λ(x,y,1)=y$ et $λ(x,x,t)=x$ for $(x,y)\in U$ and $(x,t)\in X\times[0,1]$. This class contains all ANRs, all locally contractible topological groups and the open subsets of convex subsets of linear topological spaces. In a series of papers, we extended the fixed point theory of compact continuous maps, which was well developped for ANRs, to all separeted locally equiconnected spaces. This generalization includes a proo…