arXiv subjects
Publications and source records attributed to Taras Banakh.
We explore the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse and other) defined by the size of their elements. Applying the obtained results to the Stone-Čech compactification $βG$ of $G$, we prove, in particular, that the closure of the minimal ideal of $βG$ is of type $F_{σδ}$.
We prove that a function $f:X\to Y$ from a first-countable (more generally, Preiss-Simon) space $X$ to a regular space $Y$ is weakly discontinuous (which means that every subspace $A\subset X$ contains an open dense subset $U\subset A$ such that $f|U$ is continuous) if and only if $f$ is open-resolvable (in the sense that for every open subset $U\subset Y$ the preimage $f^{-1}(U)$ is a resolvable subset of $X$) if and only if $f$ is resolvable (in the sense that for every resolvable subset $R\subset Y$ the preimage $f^{-1}(R)$ is a resolvable subset of $X$). For functions on metrizable spaces this characterization was announced (without proof) by Vinokurov in 1985.
Given a partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_α)_{α\in P}$ of subsets of $X\times X$ such that $U_β\subset U_α$ for all $α\leβ$ in $P$ and for every $x\in X$ the family $(U_α[x])_{α\in P}$ of balls $U_α[x]=\{y\in X:(x,y)\in U_α\}$ is a neighborhood base at $x$. A $P$-base $(U_α)_{α\in P}$ for $X$ is called locally uniform if the family of entourages $(U_αU_α^{-1}U_α)_{α\in P}$ remains a $P$-base for $X$. A topological space is first-countable if and only if it has an $ω$-base. By Moore's Metrization Theorem, a topological space is metrizable if and only if it is a $T_0$-space with a locally uniform $ω$-base. In the paper we shall study topological spaces possessing a (locally uniform) $ω^ω$-base. Our results show that spaces with an $ω^ω$-base share some common properties with first countable spaces, in particular, many known upper bounds on the cardinality of first-countable spaces remain true for countably tight $ω^ω$-based topological spaces. On the other hand, topological spaces with a locally uniform $ω^ω$-base have many properties, typical for generalized metric spaces. Also we study Tychonoff spaces whose universal (pre- or quasi-) uniformity has an $ω^ω$-base and show that such spaces are close to being $σ$-compact.
A topological space $X$ is defined to have an $ω^ω$-base if at each point $x\in X$ the space $X$ has a neighborhood base $(U_α[x])_{α\inω^ω}$ such that $U_β[x]\subset U_α[x]$ for all $α\leβ$ in $ω^ω$. We characterize topological and uniform spaces whose free (locally convex) topological vector spaces or free (Abelian or Boolean) topological groups have $ω^ω$-bases.
A topological space $X$ is called strongly $σ$-metrizable if $X=\bigcup_{n\inω}X_n$ for an increasing sequence $(X_n)_{n\inω}$ of closed metrizable subspaces such that every convergence sequence in $X$ is contained in some $X_n$. If, in addition, every compact subset of $X$ is contained in some $X_n$, $n\inω$, then $X$ is called super $σ$-metrizable. Answering a question of V.K.Maslyuchenko and O.I.Filipchuk, we prove that a topological space is strongly $σ$-metrizable if and only if it is super $σ$-metrizable.
A regular topological space $X$ is defined to be a $\mathfrak P_0$-space if it has countable Pytkeev network. A network $\mathcal N$ for $X$ is called a Pytkeev network if for any point $x\in X$, neighborhood $O_x\subset X$ of $x$ and subset $A\subset X$ accumulating at a $x$ there is a set $N\in\mathcal N$ such that $N\subset O_x$ and $N\cap A$ is infinite. The class of $\mathfrak P_0$-spaces contains all metrizable separable spaces and is (properly) contained in the Michael's class of $\aleph_0$-spaces. It is closed under many topological operations: taking subspaces, countable Tychonoff products, small countable box-products, countable direct limits, hyperspaces of compact subsets. For an $\aleph_0$-space $X$ and a $\mathfrak P_0$-space $Y$ the function space $C_k(X,Y)$ endowed with the compact-open topology is a $\mathfrak P_0$-space. For any sequential $\aleph_0$-space $X$ the free abelian topological group $A(X)$ and the free locally convex linear topological space $L(X)$ both are $\mathfrak P_0$-spaces. A sequential space is a $\mathfrak P_0$-space if and only if it is an $\aleph_0$-space. A topological space is metrizable and separable if and only if it is a $\mathfrak P_0$-space with countable fan tightness.
We generalize some classical results about quasicontinuous and separately continuous functions with values in metrizable spaces to functions with values in certain generalized metric spaces, called Maslyuchenko spaces. We establish stability properties of the classes of Maslyuchenko spaces and study the relation of these classes to known classes of generalized metric spaces (such as Piotrowski or Stegall spaces). One of our results says that for any $\aleph_0$-space $Z$ and any separately continuous function $f:X\times Y\to Z$ defined on the product of a topological space $X$ and a second-countable space $Y$, the set $D(f)$ of discontinuity points of $f$ has meager projection on $X$.
Modifying the known definition of a Pytkeev network, we introduce a notion of Pytkeev$^*$ network and prove that a topological space has a countable Pytkeev network if and only if $X$ is countably tight and has a countable Pykeev$^*$ network. In the paper we establish some stability properties of the class of topological spaces with the strong Pytkeev$^*$-property.
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.
Denote by $\mathbf C_k[\mathfrak M]$ the $C_k$-stable closure of the class $\mathfrak M$ of all metrizable spaces, i.e., $\mathbf C_k[\mathfrak M]$ is the smallest class of topological spaces that contains $\mathfrak M$ and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces $C_k(X,Y)$ with Lindelöf domain. We show that the class $\mathbf C_k[\mathfrak M]$ coincides with the class of all topological spaces homeomorphic to subspaces of the function spaces $C_k(X,Y)$ with a separable metrizable space $X$ and a metrizable space $Y$. We say that a topological space $Z$ is Ascoli if every compact subset of $C_k(Z)$ is evenly continuous; by the Ascoli Theorem, each $k$-space is Ascoli. We prove that the class $\mathbf C_k[\mathfrak M]$ properly contains the class of all Ascoli $\aleph_0$-spaces and is properly contained in the class of $\mathfrak P$-spaces, recently introduced by Gabriyelyan and Kąkol. Consequently, an Ascoli space $Z$ embeds into the function space $C_k(X,Y)$ for suitable separable metrizable spaces $X$ and $Y$ if and only if $Z$ is an $\aleph_0$-space.
A map $f:X\to Y$ between topological spaces is called weakly discontinuous if each subspace $A\subset X$ contains an open dense subspace $U\subset A$ such that the restriction $f|U$ is continuous. A bijective map $f:X\to Y$ between topological spaces is called a weak homeomorphism if $f$ and $f^{-1}$ are weakly discontinuous. We study properties of topological spaces preserved by weakly discontinuous maps and weak homeomorphisms. In particular, we show that weak homeomorphisms preserve network weight, hereditary Lindelöf number, dimension. Also we classify infinite zero-dimensional $σ$-Polish metrizable spaces up to a weak homeomorphism and prove that any such space $X$ is weakly homeomorphic to one of 9 spaces: $ω$, $2^ω$, $\mathbb N^ω$, $\mathbb Q$, $\mathbb Q\oplus 2^ω$, $\mathbb Q\times 2^ω$, $\mathbb Q\oplus\mathbb N^ω$, $(\mathbb Q\times 2^ω)\oplus\mathbb N^ω$, $\mathbb Q\times\mathbb N^ω$.
Answering a question of Elekes and Vidnyánszky, we construct a Polish meta-abelian group $H$ and a subgroup $F\subset H$, which is a Haar null $F_σ$-set in $H$ that cannot be enlarged to a Haar null $G_δ$-set.
For a Tychonoff space $X$, the constructions $\hat P(X)$ and $P_τ(X)$ of the spaces of probability Radon measures and probability $τ$-smooth measures on $X$ are considered. It is proved that these constructions determine functors in the category of Tychonoff spaces, which extend the functor $P$ of probability measures in the category of compacta. In this part we investigate general topological properties of the spaces $\hat P(X)$ and $P_τ(X)$, as well as categorial properties of the functors $\hat P$ and $P_τ$.
A topological group $G$ is called 2-swelling if for any compact subsets $A,B\subset G$ and elements $a,b,c\in G$ the inclusions $aA\cup bB\subset A\cup B$ and $aA\cap bB\subset c(A\cap B)$ are equivalent to the equalities $aA\cup bB=A\cup B$ and $aA\cap bB=c(A\cap B)$. We prove that an (abelian) topological group $G$ is 2-swelling if each 3-generated (resp. 2-generated) subgroup of $G$ is discrete. This implies that the additive group $\mathbb Q$ of rationals is 2-swelling and each locally finite topological group is 2-swelling.
A family of closed subsets of a topological space $X$ is called a (strict) $Cld$-fan in $X$ if this family is (strictly) compact-finite but not locally finite in $X$. Applications of (strict) $Cld$-fans are based on a simple observation that $k$-spaces contain no $Cld$-fan and Ascoli spaces contain no strict $Cld$-fan. In this paper we develop the machinery of (strict) fans and apply it to detecting the $k$-space and Ascoli properties in spaces that naturally appear in General Topology, Functional Analysis, and Topological Algebra. In particular, we detect (generalized) metric spaces $X$ whose functor-spaces, functions spaces, free (para)topological (abelian) groups, free (locally convex) linear topological spaces, free (Lawson) topological semilattices, and free (para)topological (Clifford, Abelian) inverse semigroups are $k$-spaces or Ascoli spaces.
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.
For every ballean $X$ we introduce two cardinal characteristics $cov^\flat(X)$ and $cov^\sharp(X)$ describing the capacity of balls in $X$. We observe that these cardinal characteristics are invariant under coarse equivalence and prove that two cellular ordinal balleans $X,Y$ are coarsely equivalent if $cof(X)=cof(Y)$ and $cov^\flat(X)=cov^\sharp(X)=cov^\flat(Y)=cov^\sharp(Y)$. This result implies that a cellular ordinal ballean $X$ is homogeneous if and only if $cov^\flat(X)=cov^\sharp(X)$. Moreover, two homogeneous cellular ordinal balleans $X,Y$ are coarsely equivalent if and only if $cof(X)=cof(Y)$ and $cov^\sharp(X)=cov^\sharp(Y)$ if and only if each of these balleans coarsely embeds into the other ballean. This means that the coarse structure of a homogeneous cellular ordinal ballean $X$ is fully determined by the values of the cardinals $cof(X)$ and $cov^\sharp(X)$. For every limit ordinal $γ$ we shall define a ballean $2^{<γ}$ (called the Cantor macro-cube), which in the class of cellular ordinal balleans of cofinality $cf(γ)$ plays a role analogous to the role of the Cantor cube $2^κ$ in the class of zero-dimensional compact Hausdorff spaces. We shall also present a characterization of balleans which are coarsely equivalent to $2^{<γ}$. This characterization can be considered as an asymptotic analogue of Brouwer's characterization of the Cantor cube $2^ω$.
We prove that any meager quasi-analytic subgroup of a topological group $G$ belongs to every $σ$-ideal $\mathcal I$ on $G$ possessing the closed $\pm n$-Steinhaus property for some $n\in\mathbb N$. An ideal $\mathcal I$ on a topological group $G$ is defined to have the closed $\pm n$-Steinhaus property if for any closed subsets $A_1,\dots,A_n\notin\mathcal I$ of $G$ the product $(A_1\cup A_1^{-1})\cdots (A_n\cup A_n^{-1})$ is not nowhere dense in $G$. Since the $σ$-ideal $\mathcal E$ generated by closed Haar null sets in a locally compact group $G$ has the closed $\pm 2$-Steinhaus property, we conclude that each meager quasi-analytic subgroup $H\subset G$ belongs to the ideal $\mathcal E$. For analytic subgroups of the real line this result was proved by Laczkovich in 1998. We shall discuss possible generalizations of the Laczkovich Theorem to non-locally compact groups and construct an example of a meager Borel subgroup in $\mathbb Z^ω$ which cannot be covered by countably many closed Haar-null (or even closed Haar-meager) sets. On the other hand, assuming that $cof(\mathcal M)=cov(\mathcal M)=cov(\mathcal N)$ we construct a subgroup $H\subset 2^ω$ which is meager and Haar null but does not belong to the $σ$-ideal $\mathcal E$. The construction uses a new cardinal characteristic $voc^*(\mathcal I,\mathcal J)$ which seems to be interesting by its own.