Bulletin of the Malaysian Mathematical Sciences Society, 2016
We prove that each non-metrizable sequential rectifiable space X of countable cs *-character cont... more We prove that each non-metrizable sequential rectifiable space X of countable cs *-character contains a clopen rectifiable submetrizable kω-subspace H and admits a disjoint cover by open subsets homeomorphic to clopen subspaces of H. This implies that each sequential rectifiable space of countable cs *-character is either metrizable or a topological sum of submetrizable kω-spaces. Consequently, X is submetrizable and paracompact. This answers a question of Lin and Shen posed in 2011.
For a Hausdorff topologized semilattice X its Lawson number$$\bar{\Lambda }(X)$$ Λ ¯ ( X ) is the... more For a Hausdorff topologized semilattice X its Lawson number$$\bar{\Lambda }(X)$$ Λ ¯ ( X ) is the smallest cardinal $$\kappa $$ κ such that for any distinct points $$x,y\in X$$ x , y ∈ X there exists a family $$\mathcal U$$ U of closed neighborhoods of x in X such that $$|\mathcal U|\le \kappa $$ | U | ≤ κ and $$\bigcap \mathcal U$$ ⋂ U is a subsemilattice of X that does not contain y. It follows that $$\bar{\Lambda }(X)\le \bar{\psi }(X)$$ Λ ¯ ( X ) ≤ ψ ¯ ( X ) , where $$\bar{\psi }(X)$$ ψ ¯ ( X ) is the smallest cardinal $$\kappa $$ κ such that for any point $$x\in X$$ x ∈ X there exists a family $$\mathcal U$$ U of closed neighborhoods of x in X such that $$|\mathcal U|\le \kappa $$ | U | ≤ κ and $$\bigcap \mathcal U=\{x\}$$ ⋂ U = { x } . We prove that a compact Hausdorff semitopological semilattice X is Lawson (i.e., has a base of the topology consisting of subsemilattices) if and only if $$\bar{\Lambda }(X)=1$$ Λ ¯ ( X ) = 1 . Each Hausdorff topological semilattice X has Lawson...
Let $G$ be a paratopological group.Following F.~Lin and S.~Lin, we say that the group $G$ is pseu... more Let $G$ be a paratopological group.Following F.~Lin and S.~Lin, we say that the group $G$ is pseudobounded,if for any neighborhood $U$ of the identity of $G$,there exists a natural number $n$ such that $U^n=G$.The group $G$ is $\omega$-pseudobounded,if for any neighborhood $U$ of the identity of $G$, the group $G$ is aunion of sets $U^n$, where $n$ is a natural number.The group $G$ is premeager, if $G\ne N^n$ for any nowhere dense subset $N$ of$G$ and any positive integer $n$.In this paper we investigate relations between the above classes of groups andanswer some questions posed by F. Lin, S. Lin, and S\'anchez.
A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neig... more A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-di...
We study topological spaces with the strong Skorokhod property, i.e., spaces on which all Radon p... more We study topological spaces with the strong Skorokhod property, i.e., spaces on which all Radon probability measures can be simultaneously represented as images of Lebesgue measure on the unit interval under certain Borel mappings so that weakly convergent sequences of measures correspond to almost everywhere convergent sequences of mappings. We construct nonmetrizable spaces with such a property and investigate the relations between the Skorokhod and Prokhorov properties. It is also shown that a dyadic compact has the strong Skorokhod property precisely when it is metrizable.
We construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X :... more We construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × X. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
A topological space is Suslin (Lusin) if it is a continuous (and bijective) image of a Polish spa... more A topological space is Suslin (Lusin) if it is a continuous (and bijective) image of a Polish space. For a Tychonoff space X let C p (X), C k (X) and C ↓F (X) be the space of continuous realvalued functions on X , endowed with the topology of pointwise convergence, the compactopen topology, and the Fell hypograph topology, respectively. For a metrizable space X we prove the equivalence of the following statements: (1) X is σ-compact, (2) C p (X) is Suslin, (3) C k (X) is Suslin, (4) C ↓F (X) is Suslin, (5) C p (X) is Lusin, (6) C k (X) is Lusin, (7) C ↓F (X) is Lusin, (8) C p (X) is F σ-Lusin, (9) C k (X) is F σ-Lusin, (10) C ↓F (X) is C δσ-Lusin. Also we construct an example of a sequential ℵ 0-space X with a unique non-isolated point such that the function spaces C p (X), C k (X) and C ↓F (X) are non-Suslin.
We prove that every usco multimap $$\varPhi :X\rightarrow Y$$ Φ : X → Y from a metrizable separab... more We prove that every usco multimap $$\varPhi :X\rightarrow Y$$ Φ : X → Y from a metrizable separable space X to a GO-space Y has an $$F_\sigma $$ F σ -measurable selection. On the other hand, for the split interval $${\ddot{\mathbb I}}$$ I ¨ and the projection $$P:{{\ddot{\mathbb I}}}^2\rightarrow \mathbb I^2$$ P : I ¨ 2 → I 2 of its square onto the unit square $$\mathbb I^2$$ I 2 , the usco multimap $${P^{-1}:\mathbb I^2\multimap {{\ddot{\mathbb I}}}^2}$$ P - 1 : I 2 ⊸ I ¨ 2 has a Borel ($$F_\sigma $$ F σ -measurable) selection if and only if the Continuum Hypothesis holds. This CH-example shows that know results on Borel selections of usco maps into fragmentable compact spaces cannot be extended to a wider class of compact spaces.
A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch ... more A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid S is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1.
The notion of a Banach space is one of the most fundamental notions of modern mathematics. Such s... more The notion of a Banach space is one of the most fundamental notions of modern mathematics. Such spaces were named to honour Stefan Banach (1892-1945), one of the founders of Functional Analysis, who lived, worked and died in Lviv (now the largest city in western part of Ukraine). Of course, there are many important Banach spaces: spaces of sequences, functions, operators, etc. Yet, there exists one very concrete Banach space, called the Banach space. It includes numerous historical places in Lviv related to Stefan Banach: the houses where B Taras Banakh
Let $${\mathcal {B}}$$B be a class of finite-dimensional Banach spaces. A $${\mathcal {B}}$$B-dec... more Let $${\mathcal {B}}$$B be a class of finite-dimensional Banach spaces. A $${\mathcal {B}}$$B-decomposed Banach space is a Banach space X endowed with a family $${\mathcal {B}}_X\subset {\mathcal {B}}$$BX⊂B of subspaces of X such that each $$x\in X$$x∈X can be uniquely written as the sum of an unconditionally convergent series $$\sum _{B\in {\mathcal {B}}_X}x_B$$∑B∈BXxB for some $$(x_B)_{B\in {\mathcal {B}}_X}\in \prod _{B\in {\mathcal {B}}_X}B$$(xB)B∈BX∈∏B∈BXB. For every $$B\in {\mathcal {B}}_X$$B∈BX let $$\mathrm {pr}_B:X\rightarrow B$$prB:X→B denote the coordinate projection. Let $$C\subset [-1,1]$$C⊂[-1,1] be a closed convex set with $$C\cdot C\subset C$$C·C⊂C. The C-decomposition constant $$K_C$$KC of a $${\mathcal {B}}$$B-decomposed Banach space $$(X,{\mathcal {B}}_X)$$(X,BX) is the smallest number $$K_C$$KC such that for every function $$\alpha :{\mathcal {F}}\rightarrow C$$α:F→C from a finite subset $${\mathcal {F}}\subset {\mathcal {B}}_X$$F⊂BX the operator $$T_\alpha =\sum...
Given a space $\Omega$ endowed with symmetry, we define $ms(\Omega, r)$ to be the maximum of $m$ ... more Given a space $\Omega$ endowed with symmetry, we define $ms(\Omega, r)$ to be the maximum of $m$ such that for any $r$-coloring of $\Omega$ there exists a monochromatic symmetric set of size at least $m$. We consider a wide range of spaces $\Omega$ including the discrete and continuous segments $\{1, \ldots, n\}$ and $[0,1]$ with central symmetry, geometric figures with the usual symmetries of Euclidean space, and Abelian groups with a natural notion of central symmetry. We observe that $ms(\{1, \ldots, n\}, r)$ and $ms([0,1], r)$ are closely related, prove lower and upper bounds for $ms([0,1], 2)$, and find asymptotics of $ms([0,1], r)$ for $r$ increasing. The exact value of $ms(\Omega, r)$ is determined for figures of revolution, regular polygons, and multi-dimensional parallelopipeds. We also discuss problems of a slightly different flavor and, in particular, prove that the minimal $r$ such that there exists an $r$-coloring of the $k$-dimensional integer grid without infinite mon...
Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that ... more Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set X endowed with n pairwise comparable topologies Ʈ1 ⊂ · · · ⊂ Ʈn, by repeated application of the operations of complement and closure in the topologies Ʈ1, . . . , Ʈn to a subset A ⊂ X we can obtain at most
A topologized semilattice X is called complete if each non-empty chain $$C\subset X$$ C ⊂ X has $... more A topologized semilattice X is called complete if each non-empty chain $$C\subset X$$ C ⊂ X has $$\inf C\in {\bar{C}}$$ inf C ∈ C ¯ and $$\sup C\in {\bar{C}}$$ sup C ∈ C ¯ . We prove that for any continuous homomorphism $$h:X\rightarrow Y$$ h : X → Y from a complete topologized semilattice X to a sequential Hausdorff semitopological semilattice Y the image h(X) is closed in Y.
We survey some results and pose some open problems related to boundedness of real-valued function... more We survey some results and pose some open problems related to boundedness of real-valued functions on balleans and coarse spaces. Special attention is paid to balleans on groups. The boundedness of functions that respect the coarse structure of a ballean could be considered as a coarse counterpart of pseudo-compactness.
Bulletin of the Malaysian Mathematical Sciences Society, 2016
We prove that each non-metrizable sequential rectifiable space X of countable cs *-character cont... more We prove that each non-metrizable sequential rectifiable space X of countable cs *-character contains a clopen rectifiable submetrizable kω-subspace H and admits a disjoint cover by open subsets homeomorphic to clopen subspaces of H. This implies that each sequential rectifiable space of countable cs *-character is either metrizable or a topological sum of submetrizable kω-spaces. Consequently, X is submetrizable and paracompact. This answers a question of Lin and Shen posed in 2011.
For a Hausdorff topologized semilattice X its Lawson number$$\bar{\Lambda }(X)$$ Λ ¯ ( X ) is the... more For a Hausdorff topologized semilattice X its Lawson number$$\bar{\Lambda }(X)$$ Λ ¯ ( X ) is the smallest cardinal $$\kappa $$ κ such that for any distinct points $$x,y\in X$$ x , y ∈ X there exists a family $$\mathcal U$$ U of closed neighborhoods of x in X such that $$|\mathcal U|\le \kappa $$ | U | ≤ κ and $$\bigcap \mathcal U$$ ⋂ U is a subsemilattice of X that does not contain y. It follows that $$\bar{\Lambda }(X)\le \bar{\psi }(X)$$ Λ ¯ ( X ) ≤ ψ ¯ ( X ) , where $$\bar{\psi }(X)$$ ψ ¯ ( X ) is the smallest cardinal $$\kappa $$ κ such that for any point $$x\in X$$ x ∈ X there exists a family $$\mathcal U$$ U of closed neighborhoods of x in X such that $$|\mathcal U|\le \kappa $$ | U | ≤ κ and $$\bigcap \mathcal U=\{x\}$$ ⋂ U = { x } . We prove that a compact Hausdorff semitopological semilattice X is Lawson (i.e., has a base of the topology consisting of subsemilattices) if and only if $$\bar{\Lambda }(X)=1$$ Λ ¯ ( X ) = 1 . Each Hausdorff topological semilattice X has Lawson...
Let $G$ be a paratopological group.Following F.~Lin and S.~Lin, we say that the group $G$ is pseu... more Let $G$ be a paratopological group.Following F.~Lin and S.~Lin, we say that the group $G$ is pseudobounded,if for any neighborhood $U$ of the identity of $G$,there exists a natural number $n$ such that $U^n=G$.The group $G$ is $\omega$-pseudobounded,if for any neighborhood $U$ of the identity of $G$, the group $G$ is aunion of sets $U^n$, where $n$ is a natural number.The group $G$ is premeager, if $G\ne N^n$ for any nowhere dense subset $N$ of$G$ and any positive integer $n$.In this paper we investigate relations between the above classes of groups andanswer some questions posed by F. Lin, S. Lin, and S\'anchez.
A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neig... more A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-di...
We study topological spaces with the strong Skorokhod property, i.e., spaces on which all Radon p... more We study topological spaces with the strong Skorokhod property, i.e., spaces on which all Radon probability measures can be simultaneously represented as images of Lebesgue measure on the unit interval under certain Borel mappings so that weakly convergent sequences of measures correspond to almost everywhere convergent sequences of mappings. We construct nonmetrizable spaces with such a property and investigate the relations between the Skorokhod and Prokhorov properties. It is also shown that a dyadic compact has the strong Skorokhod property precisely when it is metrizable.
We construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X :... more We construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × X. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
A topological space is Suslin (Lusin) if it is a continuous (and bijective) image of a Polish spa... more A topological space is Suslin (Lusin) if it is a continuous (and bijective) image of a Polish space. For a Tychonoff space X let C p (X), C k (X) and C ↓F (X) be the space of continuous realvalued functions on X , endowed with the topology of pointwise convergence, the compactopen topology, and the Fell hypograph topology, respectively. For a metrizable space X we prove the equivalence of the following statements: (1) X is σ-compact, (2) C p (X) is Suslin, (3) C k (X) is Suslin, (4) C ↓F (X) is Suslin, (5) C p (X) is Lusin, (6) C k (X) is Lusin, (7) C ↓F (X) is Lusin, (8) C p (X) is F σ-Lusin, (9) C k (X) is F σ-Lusin, (10) C ↓F (X) is C δσ-Lusin. Also we construct an example of a sequential ℵ 0-space X with a unique non-isolated point such that the function spaces C p (X), C k (X) and C ↓F (X) are non-Suslin.
We prove that every usco multimap $$\varPhi :X\rightarrow Y$$ Φ : X → Y from a metrizable separab... more We prove that every usco multimap $$\varPhi :X\rightarrow Y$$ Φ : X → Y from a metrizable separable space X to a GO-space Y has an $$F_\sigma $$ F σ -measurable selection. On the other hand, for the split interval $${\ddot{\mathbb I}}$$ I ¨ and the projection $$P:{{\ddot{\mathbb I}}}^2\rightarrow \mathbb I^2$$ P : I ¨ 2 → I 2 of its square onto the unit square $$\mathbb I^2$$ I 2 , the usco multimap $${P^{-1}:\mathbb I^2\multimap {{\ddot{\mathbb I}}}^2}$$ P - 1 : I 2 ⊸ I ¨ 2 has a Borel ($$F_\sigma $$ F σ -measurable) selection if and only if the Continuum Hypothesis holds. This CH-example shows that know results on Borel selections of usco maps into fragmentable compact spaces cannot be extended to a wider class of compact spaces.
A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch ... more A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid S is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1.
The notion of a Banach space is one of the most fundamental notions of modern mathematics. Such s... more The notion of a Banach space is one of the most fundamental notions of modern mathematics. Such spaces were named to honour Stefan Banach (1892-1945), one of the founders of Functional Analysis, who lived, worked and died in Lviv (now the largest city in western part of Ukraine). Of course, there are many important Banach spaces: spaces of sequences, functions, operators, etc. Yet, there exists one very concrete Banach space, called the Banach space. It includes numerous historical places in Lviv related to Stefan Banach: the houses where B Taras Banakh
Let $${\mathcal {B}}$$B be a class of finite-dimensional Banach spaces. A $${\mathcal {B}}$$B-dec... more Let $${\mathcal {B}}$$B be a class of finite-dimensional Banach spaces. A $${\mathcal {B}}$$B-decomposed Banach space is a Banach space X endowed with a family $${\mathcal {B}}_X\subset {\mathcal {B}}$$BX⊂B of subspaces of X such that each $$x\in X$$x∈X can be uniquely written as the sum of an unconditionally convergent series $$\sum _{B\in {\mathcal {B}}_X}x_B$$∑B∈BXxB for some $$(x_B)_{B\in {\mathcal {B}}_X}\in \prod _{B\in {\mathcal {B}}_X}B$$(xB)B∈BX∈∏B∈BXB. For every $$B\in {\mathcal {B}}_X$$B∈BX let $$\mathrm {pr}_B:X\rightarrow B$$prB:X→B denote the coordinate projection. Let $$C\subset [-1,1]$$C⊂[-1,1] be a closed convex set with $$C\cdot C\subset C$$C·C⊂C. The C-decomposition constant $$K_C$$KC of a $${\mathcal {B}}$$B-decomposed Banach space $$(X,{\mathcal {B}}_X)$$(X,BX) is the smallest number $$K_C$$KC such that for every function $$\alpha :{\mathcal {F}}\rightarrow C$$α:F→C from a finite subset $${\mathcal {F}}\subset {\mathcal {B}}_X$$F⊂BX the operator $$T_\alpha =\sum...
Given a space $\Omega$ endowed with symmetry, we define $ms(\Omega, r)$ to be the maximum of $m$ ... more Given a space $\Omega$ endowed with symmetry, we define $ms(\Omega, r)$ to be the maximum of $m$ such that for any $r$-coloring of $\Omega$ there exists a monochromatic symmetric set of size at least $m$. We consider a wide range of spaces $\Omega$ including the discrete and continuous segments $\{1, \ldots, n\}$ and $[0,1]$ with central symmetry, geometric figures with the usual symmetries of Euclidean space, and Abelian groups with a natural notion of central symmetry. We observe that $ms(\{1, \ldots, n\}, r)$ and $ms([0,1], r)$ are closely related, prove lower and upper bounds for $ms([0,1], 2)$, and find asymptotics of $ms([0,1], r)$ for $r$ increasing. The exact value of $ms(\Omega, r)$ is determined for figures of revolution, regular polygons, and multi-dimensional parallelopipeds. We also discuss problems of a slightly different flavor and, in particular, prove that the minimal $r$ such that there exists an $r$-coloring of the $k$-dimensional integer grid without infinite mon...
Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that ... more Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set X endowed with n pairwise comparable topologies Ʈ1 ⊂ · · · ⊂ Ʈn, by repeated application of the operations of complement and closure in the topologies Ʈ1, . . . , Ʈn to a subset A ⊂ X we can obtain at most
A topologized semilattice X is called complete if each non-empty chain $$C\subset X$$ C ⊂ X has $... more A topologized semilattice X is called complete if each non-empty chain $$C\subset X$$ C ⊂ X has $$\inf C\in {\bar{C}}$$ inf C ∈ C ¯ and $$\sup C\in {\bar{C}}$$ sup C ∈ C ¯ . We prove that for any continuous homomorphism $$h:X\rightarrow Y$$ h : X → Y from a complete topologized semilattice X to a sequential Hausdorff semitopological semilattice Y the image h(X) is closed in Y.
We survey some results and pose some open problems related to boundedness of real-valued function... more We survey some results and pose some open problems related to boundedness of real-valued functions on balleans and coarse spaces. Special attention is paid to balleans on groups. The boundedness of functions that respect the coarse structure of a ballean could be considered as a coarse counterpart of pseudo-compactness.
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