We answer a question of Piotr Minc by proving that there is no compact metrizable space whose set of components contains a unique topological copy of every metrizable compactification of a ray (i.e. a half-open interval) with an arc (i.e. closed bounded interval) as the remainder. To this end we use the concept of Borel reductions coming from Invariant descriptive set theory. It follows as a corollary that there is no compact metrizable space such that every continuum is homeomorphic to exactly one component of this space.
Fedorchuk's fully closed (continuous) maps and resolutions are applied in constructions of non-metrizable higher-dimensional analogues of Anderson, Choquet, and Cook's continua. Certain theorems on dimension-lowering maps are proved for inductive dimensions and fully closed maps from spaces that need not be hereditarily normal, and some examples of continua have non-coinciding dimensions.
We prove that, for any Hausdorff continuum X, if dim X > 1 then the hyperspace C(X) of subcontinua of X is not a C-space; if dim X = 1 and X is hereditarily indecomposable then dim C(X) = 2 or C(X) is not a C-space. This generalizes results known for metric continua.
We show that if a separable space X has a meager open subset containing a copy of the Cantor set 2^ω, then X has $\frak{c}$ types of countable dense subsets. We suggest a generalization of the λ-set for non-separable spaces. Let X be an h-homogeneous Λ-set. Then X is densely homogeneous and (X \setminus A) is homeomorphic to X for every σ-discrete A \subset X.
Suppose G is a topological group containing a (closed) topological copy of the Frechet-Urysohn fan. If G is a perfectly normal sequential space (a normal k-space) then every closed metrizable subset in $G$ is locally compact. Applying this result to topological groups whose underlying topological space can be written as a direct limit of a sequence of closed metrizable subsets, we get that every such a group either is metrizable or is homeomorphic to the product of a $k_ω$-space and a discrete space.
A Hausdorff topological group topology on a group $G$ is the minimum (Hausdorff) group topology if it is contained in every Hausdorff group topology on $G$. For every compact metrizable space $X$ containing an open $n$-cell, $n\ge2$, the homeomorphism group $H(X)$ has no minimum Hausdorff group topology. The homeomorphism groups of the Cantor set and the Hilbert cube have no minimum group topology. For every compact metrizable space $X$ containing a dense open one-manifold, $H(X)$ has the minimum group topology. Some, but not all, oligomorphic groups have the minimum group topology.
We prove that the road space of an R-special tree is contractible and that a locally metrizable space containing a copy of an uncountable $ω_1$-compact subspace of a tree is not. We also raise some questions about possible generalizations.
The deck of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}] \colon x \in X\}$, where $[Z]$ denotes the homeomorphism class of $Z$. A space $X$ is topologically reconstructible if whenever $\mathcal{D}(X)=\mathcal{D}(Y)$ then $X$ is homeomorphic to $Y$. It is shown that all metrizable compact connected spaces are reconstructible. It follows that all finite graphs, when viewed as a 1-dimensional cell-complex, are reconstructible in the topological sense, and more generally, that all compact graph-like spaces are reconstructible.
Let $Y$ be a metrizable space containing at least two points, and let $X$ be a $Y_{\mathcal{I}}$-Tychonoff space for some ideal $\mathcal{I}$ of compact sets of $X$. Denote by $C_{\mathcal{I}}(X,Y)$ the space of continuous functions from $X$ to $Y$ endowed with the $\mathcal{I}$-open topology. We prove that $C_{\mathcal{I}}(X,Y)$ is Fréchet - Urysohn iff $X$ has the property $γ_{\mathcal{I}}$. We characterize zero - dimensional Tychonoff spaces $X$ for which the space $C_{\mathcal{I}}(X,{\bf 2})$ is sequential. Extending the classical theorems of Gerlits, Nagy and Pytkeev we show that if $Y$ i…
A locally convex space (lcs) $E$ is said to have an $ω^ω$-base if $E$ has a neighborhood base $\{U_α:α\inω^ω\}$ at zero such that $U_β\subseteq U_α$ for all $α\leqβ$. The class of lcs with an $ω^ω$-base is large, among others contains all $(LM)$-spaces (hence $(LF)$-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $D'(Ω)$). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an $ω^ω$-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $ω^ω$-base contains an infinite-dimensional metrizable com…
A compact graph-like space is a triple $(X,V,E)$ where $X$ is a compact, metrizable space, $V \subseteq X$ is a closed zero-dimensional subset, and $E$ is an index set such that $X \setminus V \cong E \times (0,1)$. New characterizations of compact graph-like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph-like compacta.
We show that for every $d\ge 1$, if $L_1,\ldots, L_d$ are linearly ordered compact spaces and there is a continuous surjection \[ L_1\times L_2\times \dots\times L_d\to K_1\times K_2\times\ldots\times K_{d}\times K_{d+1},\] where all the spaces $K_i$ are infinite, then $K_i, K_j$ are metrizable for some $1\le i<j\le d+1$. This answers a problem posed by S. Mardešić. We present some related results on Boolean algebras not containing free products with too many uncountable factors. In particular, we answer a problem on initial chain algebras that was posed by L. Baur and L. Heindorf.
Given a continuum $X$ and $p\in X$, we will consider the hyperspace $C(p,X)$ of all subcontinua of $X$ containing $p$. Given a family of continua $\mathcal{C}$, a continuum $X\in\mathcal{C}$ and $p\in X$, we say that $(X,p)$ has unique hyperspace $C(p,X)$ relative to $\mathcal{C}$ if for each $Y\in\mathcal{C}$ and $q\in Y$ such that $C(p,X)$ and $C(q,Y)$ are homeomorphic, then there is an homeomorphism between $X$ and $Y$ sending $p$ to $q$. In this paper we show that $(X,p)$ has unique hyperspace $C(p,X)$ relative to the classes of dendrites if and only if $X$ is a tree, we present also some …
N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences（俄罗斯科学院乌拉尔分校 N.N.克拉索夫斯基数学力学研究所）；Department of General Topology and Geometry, Mechanics and Mathematics Faculty, M. V. Lomonosov Moscow State University（莫斯科国立大学机械数学系一般拓扑与几何教研室）
We define the compact universal cover of a compact, metrizable connected space (i.e. a continuum) X to be the inverse limit of all continua that regularly cover X. We show that such covers do indeed form an inverse system with bonding maps that are regular covering maps, and the projection from the inverse limit is a generalized regular covering map in the sense of Berestovskii-Plaut. The inverse limit space is a continuum that is "compactly simply connected" in the sense that its "profinite fundamental group" (the inverse limit of the deck groups of the finite covers) is trivial. We prove a G…
We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of $l_1$. For that purpose, we transfer to general locally compact groups the notion of interpolation ($I_0$) set, which was defined by Hartman and Ryll-Nardzewsky [25] for locally compact abelian groups. Thus we prove that for every sequence $\lbrace g_n \rbrace_{n<ω}$ in a locally compact group $G$, then either $\lbrace g_n \rbrace_{n<ω}$ has a weak Cauchy subsequence or contains a subsequence that is an $I_0$ set. This result is subsequently applied to obtain sufficient condi…
Given a class $\mathcal P$ of Banach spaces, a locally convex space (LCS) $E$ is called {\em multi-$\mathcal P$} if $E$ can be isomorphically embedded into a product of spaces that belong to $\mathcal P$. We investigate the question whether the free locally convex space $L(X)$ is strongly nuclear, nuclear, Schwartz, multi-Hilbert or multi-reflexive. If $X$ is a Tychonoff space containing an infinite compact subset then, as it follows from the results of \cite{Aus}, $L(X)$ is not nuclear. We prove that for such $X$ the free LCS $L(X)$ has the stronger property of not being multi-Hilbert. We ded…
