arXiv subjects
Publications and source records attributed to Vladimir Uspenskij.
We answer the question, raised more than thirty years ago, on whether the power (G raised to the power omega) of a countably compact minimal Abelian group G is minimal, by showing that the negative answer is equivalent to the existence of measurable cardinals.
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 deduce that if $X$ is a $k$-space, then the following properties are equivalent: (1) $L(X)$ is strongly nuclear; (2) $L(X)$ is nuclear; (3) $L(X)$ is multi-Hilbert; (4) $X$ is countable and discrete. On the other hand, we show that $L(X)$ is strongly nuclear for every projectively countable $P$-space (in particular, for every Lindelöf $P$-space) $X$. We observe that every Schwartz LCS is multi-reflexive. It is known that if $X$ is a $k_ω$-space, then $L(X)$ is a Schwartz LCS \cite{Chasco}, hence $L(X)$ is multi-reflexive. We show that for any first-countable paracompact (in particular, metrizable) space $X$ the converse is true, so $L(X)$ is multi-reflexive if and only if $X$ is a $k_ω$-space, equivalently, if $X$ is a locally compact and $σ$-compact space. Similarly, we show that for any first-countable paracompact space $X$ the free abelian topological group $A(X)$ is a Schwartz group if and only if $X$ is a locally compact space such that the set $X^{(1)}$ of all non-isolated points of $X$ is $σ$-compact.
A.V.Arkhangel'skii asked in 1981 if the variety $\mathfrak V$ of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued measurable cardinal then the variety $\mathfrak V$ is a proper subclass of the class of all topological groups. A topological group $G$ is called $g$-sequential if for any topological group $H$ any sequentially continuous homomorphism $G\to H$ is continuous. We introduce the concept of a $g$-sequential cardinal and prove that a locally compact group is $g$-sequential if and only if its local weight is not a $g$-sequential cardinal. The product of a family of non-trivial $g$-sequential topological groups is $g$-sequential if and only if the cardinal of this family is not $g$-sequential. Suppose $G$ is either the unitary group of a Hilbert space or the group of all self-homeomorphisms of a Tikhonov cube. Then $G$ is $g$-sequential if and only if its weight is not a $g$-sequential cardinal. Every compact group of Ulam-measurable cardinality admits a strictly finer countably compact group topology.
A topological group $G$ is {\em extremely amenable} if every compact $G$-space has a $G$-fixed point. Let $X$ be compact and $G\subset{\mathrm{Homeo}} (X)$. We prove that the following are equivalent: (1) $G$ is extremely amenable; (2) every minimal closed $G$-invariant subset of $\exp R$ is a singleton, where $R$ is the closure of the set of all graphs of $g\in G$ in the space $\exp (X^2)$ ($\exp$ stands for the space of closed subsets); (3) for each $n=1,2,...$ there is a closed $G$-invariant subset $Y_n$ of $(\exp X)^n$ such that $\cup_{n=1}^\infty Y_n$ contains arbitrarily fine covers of $X$ and for every $n\ge 1$ every minimal closed $G$-invariant subset of $\exp Y_n$ is a singleton. This yields an alternative proof of Pestov's theorem that the group of all order-preserving self-homeomorphisms of the Cantor middle-third set (or of the interval $[0,1]$) is extremely amenable.
Suppose that a topological space $X$ is the union of an increasing sequence of open subsets each of which is homeomorphic to the Euclidean space $R^n$. Then $X$ itself is homeomorphic to $R^n$. This is an old theorem of Morton Brown. We observe that this theorem is an immediate consequence of other two theorems of Morton Brown concerning near homeomorphisms and cellular sets.
A compact space $K$ is {\em Valdivia compact} if it can be embedded in a Tikhonov cube $I^A$ in such a way that the intersection $K\capΣ$ is dense in $K$, where $Σ$ is the sigma-product (= the set of points with countably many non-zero coordinates). We show that there exists a compact connected Abelian group of weight $ø_1$ which is not Valdivia compact, and deduce that Valdivia compact spaces are not preserved by open maps.
The Urysohn universal metric space U is characterized up to isometry by the following properties: (1) U is complete and separable; (2) U contains an isometric copy of every separable metric space; (3) every isometry between two finite subsets of U can be extended to an isometry of U onto itself. We show that U is homeomorphic to the Hilbert space l_2 (or to the countable power of the real line).
Every topological group $G$ has some natural compactifications which can be a useful tool of studying $G$. We discuss the following constructions: (1) the greatest ambit $S(G)$ is the compactification corresponding to the algebra of all right uniformly continuous bounded functions on $G$; (2) the Roelcke compactification $R(G)$ corresponds to the algebra of functions which are both left and right uniformly continuous; (3) the weakly almost periodic compactification $W(G)$ is the envelopping compact semitopological semigroup of $G$ (`semitopological' means that the multiplication is separately continuous). The universal minimal compact $G$-space $X=M_G$ is characterized by the following properties: (1) $X$ has no proper closed $G$-invariant subsets; (2) for every compact $G$-space $Y$ there exists a $G$-map $X\to Y$. A group $G$ is extremely amenable, or has the fixed point on compacta property, if $M_G$ is a singleton. We discuss some results and questions by V. Pestov and E. Glasner on extremely amenable groups. The Roelcke compactifications were used by M. Megrelishvili to prove that $W(G)$ can be a singleton. They can be used to prove that certain groups are minimal. A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology.
For every topological group G one can define the universal minimal compact G-space X=M_G characterized by the following properties: (1) X has no proper closed G-invariant subsets; (2) for every compact G-space Y there exists a G-map X-->Y. If G is the group of all orientation-preserving homeomorphisms of the circle S^1, then M_G can be identified with S^1 (V. Pestov). We show that the circle cannot be replaced by the Hilbert cube or a compact manifold of dimension >1. This answers a question of V. Pestov. Moreover, we prove that for every topological group G the action of G on M_G is not 3-transitive.