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Saak Gabriyelyan

Publications and source records attributed to Saak Gabriyelyan.

At least 37 records · Page 2Linked to original sources
arXiv2021

A simple Efimov space with sequentially-nice space of probability measures

Under Jensen's diamond principle $\diamondsuit$, we construct a simple Efimov space $K$ whose space of nonatomic probability measures $P_{na}(K)$ is first-countable and sequentially compact. These two properties of $P_{na}(K)$ imply that the space of probability measures $P(K)$ on $K$ is selectively sequentially pseudocompact and the Banach space $C(K)$ of continuous functions on $K$ has the Gelfand-Phillips property. We show also that any sequence of probability measures on $K$ that converges to an atomic measure converges in norm, and any sequence of probability measures on $K$ converging to zero in sup-norm has a subsequence converging to a nonatomic probability measure.

math.GN↗
arXiv2021

Banach spaces with the (strong) Gelfand--Phillips property

Several new characterizations of the Gelfand-Phillips property are given. We define a strong version of the Gelfand-Phillips property and prove that a Banach space has this stronger property iff it embeds into $c_0$. For an infinite compact space $K$, the Banach space $C(K)$ has the strong Gelfand-Phillips property iff $C(K)$ is isomorphic to $c_0$ iff $K$ is countable and has finite scattered height.

math.FA↗
arXiv2020

Ascoli's theorem for pseudocompact spaces

A Tychonoff space $X$ is called ({\em sequentially}) {\em Ascoli} if every compact subset (resp. convergent sequence) of $C_k(X)$ is equicontinuous, where $C_k(X)$ denotes the space of all real-valued continuous functions on $X$ endowed with the compact-open topology. The classical Ascoli theorem states that each compact space is Ascoli. We show that a pseudocompact space $X$ is Asoli iff it is sequentially Ascoli iff it is selectively $ω$-bounded.

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arXiv2020

Topological properties of some function spaces

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 $Y$ is not compact, then $C_{p}(X,Y)$ is Fréchet - Urysohn iff it is sequential iff it is a $k$-space iff $X$ has the property $γ$. An analogous result is obtained for the space of bounded continuous functions taking values in a metrizable locally convex space. Denote by $B_{1}(X,Y)$ and $B(X,Y)$ the space of Baire one functions and the space of all Baire functions from $X$ to $Y$, respectively. If $H$ is a subspace of $B(X,Y)$ containing $B_{1}(X,Y)$, then $H$ is metrizable iff it is a $σ$ - space iff it has countable $cs^*$ - character iff $X$ is countable. If additionally $Y$ is not compact, then $H$ is Fréchet - Urysohn iff it is sequential iff it is a $k$ - space iff it has countable tightness iff $X_{\aleph_0}$ has the property $γ$, where $X_{\aleph_0}$ is the space $X$ with the Baire topology. We show that if $X$ is a Polish space, then the space $B_{1}(X,\mathbb{R})$ is normal iff $X$ is countable.

math.GN↗
arXiv2020

Ascoli and sequentially Ascoli spaces

A Tychonoff space $X$ is called ({\em sequentially}) {\em Ascoli} if every compact subset (resp. convergent sequence) of $C_k(X)$ is evenly continuous, where $C_k(X)$ denotes the space of all real-valued continuous functions on $X$ endowed with the compact-open topology. Various properties of (sequentially) Ascoli spaces are studied, and we give several characterizations of sequentially Ascoli spaces. Strengthening a result of Arhangel'skii we show that a hereditary Ascoli space is Fréchet--Urysohn. A locally compact abelian group $G$ with the Bohr topology is sequentially Ascoli iff $G$ is compact. If $X$ is totally countably compact or near sequentially compact then it is a sequentially Ascoli space. The product of a locally compact space and an Ascoli space is Ascoli. If additionally $X$ is a $μ$-space, then $X$ is locally compact iff the product of $X$ with any Ascoli space is an Ascoli space. Extending one of the main results of [18] and [16] we show that $C_p(X)$ is sequentially Ascoli iff $X$ has the property $(κ)$. We give a necessary condition on $X$ for which the space $C_k(X)$ is sequentially Ascoli. For every metrizable abelian group $Y$, $Y$-Tychonoff space $X$, and nonzero countable ordinal $α$, the space $B_α(X,Y)$ of Baire-$α$ functions from $X$ to $Y$ is $κ$-Fréchet--Urysohn and hence Ascoli.

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arXiv2019

Topological properties of spaces of Baire functions

A fundamental result proved by Bourgain, Fremlin and Talagrand states that the space $B_1(M)$ of Baire one functions over a Polish space $M$ is an angelic space. Stegall extended this result by showing that the class $B_1(M,E)$ of Baire one functions valued in a normed space $E$ is angelic. These results motivate our study of various topological properties in the classes $B_α(X,G)$ of Baire-$α$ functions, where $α$ is a nonzero countable ordinal, $G$ is a metrizable non-precompact abelian group and $X$ is a $G$-Tychonoff first countable space. In particular, we show that (1) $B_α(X,G)$ is a $κ$-Fréchet--Urysohn space and hence it is an Ascoli space, and (2) $B_α(X,G)$ is a $k$-space iff $X$ is countable.

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arXiv2018

Locally convex properties of free locally convex spaces

Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. We show that the following assertions are equivalent: (i) $L(X)$ is $\ell_\infty$-barrelled, (ii) $L(X)$ is $\ell_\infty$-quasibarrelled, (iii) $L(X)$ is $c_0$-barrelled, (iv) $L(X)$ is $\aleph_0$-quasibarrelled, and (v) $X$ is a $P$-space. If $X$ is a non-discrete metrizable space, then $L(X)$ is $c_0$-quasibarrelled but it is neither $c_0$-barrelled nor $\ell_\infty$-quasibarrelled. We prove that $L(X)$ is a $(DF)$-space iff $X$ is a countable discrete space. We show that there is a countable Tychonoff space $X$ such that $L(X)$ is a quasi-$(DF)$-space but is not a $c_0$-quasibarrelled space. For each non-metrizable compact space $K$, the space $L(K)$ is a $(df)$-space but is not a quasi-$(DF)$-space. If $X$ is a $μ$-space, then $L(X)$ has the Grothendieck property iff every compact subset of $X$ is finite. We show that $L(X)$ has the Dunford--Pettis property for every Tychonoff space $X$. If $X$ is a sequential $μ$-space (for example, metrizable), then $L(X)$ has the sequential Dunford--Pettis property iff $X$ is discrete.

math.FA↗
arXiv2018

Locally convex spaces and Schur type properties

In the main result of the paper we extend Rosenthal's characterization of Banach spaces with the Schur property by showing that for a quasi-complete locally convex space $E$ whose separable bounded sets are metrizable the following conditions are equivalent: (1) $E$ has the Schur property, (2) $E$ and $E_w$ have the same sequentially compact sets, where $E_w$ is the space $E$ with the weak topology, (3) $E$ and $E_w$ have the same compact sets, (4) $E$ and $E_w$ have the same countably compact sets, (5) $E$ and $E_w$ have the same pseudocompact sets, (6) $E$ and $E_w$ have the same functionally bounded sets, (7) every bounded non-precompact sequence in $E$ has a subsequence which is equivalent to the unit basis of $\ell_1$ and (8) every bounded non-precompact sequence in $E$ has a subsequence which is discrete and $C$-embedded in $E_w$.

math.FA↗
arXiv2018

Dunford--Pettis type properties and the Grothendieck property for function spaces

For a Tychonoff space $X$, let $C_k(X)$ and $C_p(X)$ be the spaces of real-valued continuous functions $C(X)$ on $X$ endowed with the compact-open topology and the pointwise topology, respectively. If $X$ is compact, the classic result of A.~Grothendieck states that $C_k(X)$ has the Dunford-Pettis property and the sequential Dunford--Pettis property. We extend Grothendieck's result by showing that $C_k(X)$ has both the Dunford-Pettis property and the sequential Dunford-Pettis property if $X$ satisfies one of the following conditions: (i) $X$ is a hemicompact space, (ii) $X$ is a cosmic space (=a continuous image of a separable metrizable space), (iii) $X$ is the ordinal space $[0,κ)$ for some ordinal $κ$, or (vi) $X$ is a locally compact paracompact space. We show that if $X$ is a cosmic space, then $C_k(X)$ has the Grothendieck property if and only if every functionally bounded subset of $X$ is finite. We prove that $C_p(X)$ has the Dunford--Pettis property and the sequential Dunford-Pettis property for every Tychonoff space $X$, and $C_p(X) $ has the Grothendieck property if and only if every functionally bounded subset of $X$ is finite.

math.FA↗
arXiv2018

On reflexivity and the Ascoli property for free locally convex spaces

Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. If $X$ is Dieudonné complete (for example, metrizable), then $L(X)$ is a reflexive group if and only if $X$ is discrete. Answering a question posed in [9] we prove also that $L(X)$ is an Ascoli space if and only if $X$ is a countable discrete space.

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arXiv2018

Topological properties of inductive limits of closed towers of mertrizable groups

Let $\{ G_n\}_{n\in\w}$ be a closed tower of metrizable groups. Under a mild condition called $(GC)$ and which is strictly weaker than $PTA$ condition introduced in [22], we show that: (1) the inductive limit $G=\mbox{g-}\underrightarrow{\lim}\, G_n$ of the tower is a Hausdorff group, (2) every $G_n$ is a closed subgroup of $G$, (3) if $K$ is a compact subset of $G$, then $K\subseteq G_m$ for some $m\inω$, (4) $G$ has a $\mathfrak{G}$-base and countable tightness, (5) $G$ is an $\aleph$-space, (6) $G$ is an Ascoli space if and only if either (i) there is $m\inω$ such that $G_n$ is open in $G_{n+1}$ for every $n\geq m$, so $G$ is metrizable; or (ii) all the groups $G_n$ are locally compact and $G$ is a sequential non-Fréchet--Urysohn space.

math.GN↗
arXiv2018

Maximally almost periodic groups and respecting properties

For a Tychonoff space $X$, denote by $\mathfrak{P}$ the family of topological properties $\mathcal{P}$ of being a convergent sequence or being a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of $X$, respectively. A maximally almost periodic $(MAP$) group $G$ respects $\mathcal{P}$ if $\mathcal{P}(G)=\mathcal{P}(G^+)$, where $G^+$ is the group $G$ endowed with the Bohr topology. We study relations between different respecting properties from $\mathfrak{P}$ and show that the respecting convergent sequences (=the Schur property) is the weakest one among the properties of $\mathfrak{P}$. We characterize respecting properties from $\mathfrak{P}$ in wide classes of $MAP$ topological groups including the class of metrizable $MAP$ abelian groups. Every real locally convex space (lcs) is a quotient space of an lcs with the Schur property, and every locally quasi-convex (lqc) abelian group is a quotient group of an lqc abelian group with the Schur property. It is shown that a reflexive group $G$ has the Schur property or respects compactness iff its dual group $G^\wedge$ is $c_0$-barrelled or $g$-barrelled, respectively. We prove that a locally quasi-convex abelian $k_ω$-group respects all properties $\mathcal{P}\in\mathfrak{P}$. As an application of the obtained results we show that (1) the space $C_k(X)$ is a reflexive group for every separable metrizable space $X$, and (2) a reflexive abelian group of finite exponent is a Mackey group.

math.GN↗
arXiv2018

On the Mackey problem for free abelian topological groups

Recently Außenhofer and the author independently have shown that the free abelian topological group $A(\mathbf{s})$ over a convergent sequence $\mathbf{s}$ does not admit the strongest compatible locally quasi-convex group topology that gives the first example of a locally quasi-convex abelian group without a Mackey group topology. In this note we considerably extend this example by showing that the free abelian topological group $A(X)$ over a non-discrete zero-dimensional metrizable space $X$ does not have a Mackey group topology. In particular, for every countable non-discrete metrizable space $X$, the group $A(X)$ does not have a Mackey group topology.

math.GN↗
arXiv2017

A locally quasi-convex abelian group without Mackey topology

We give the first example of a locally quasi-convex (even countable reflexive and $k_ω$) abelian group $G$ which does not admit the strongest compatible locally quasi-convex group topology. Our group $G$ is the Graev free abelian group $A_G(\mathbf{s})$ over a convergent sequence $\mathbf{s}$.

math.GN↗
arXiv2017

Topological properties of strict $(LF)$-spaces and strong duals of Montel strict $(LF)$-spaces

Following [2], a Tychonoff space $X$ is Ascoli if every compact subset of $C_k(X)$ is equicontinuous. By the classical Ascoli theorem every $k$-space is Ascoli. We show that a strict $(LF)$-space $E$ is Ascoli iff $E$ is a Fréchet space or $E=ϕ$. We prove that the strong dual $E'_β$ of a Montel strict $(LF)$-space $E$ is an Ascoli space iff one of the following assertions holds: (i) $E$ is a Fréchet--Montel space, so $E'_β$ is a sequential non-Fréchet--Urysohn space, or (ii) $E=ϕ$, so $E'_β= \mathbb{R}^ω$. Consequently, the space $\mathcal{D}(Ω)$ of test functions and the space of distributions $\mathcal{D}'(Ω)$ are not Ascoli that strengthens results of Shirai [20] and Dudley [5], respectively.

math.FA↗
arXiv2016

Topological properties of function spaces over ordinals

A topological space $X$ is said to be an Ascoli space if any compact subset $K$ of $C_k(X)$ is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the $k_R$-property and the Ascoli property of $C_p(κ)$ and $C_k(κ)$ over ordinals $κ$. We prove that $C_p(κ)$ is always an Ascoli space, while $C_p(κ)$ is a $k_R$-space iff the cofinality of $κ$ is countable. In particular, this provides the first $C_p$-example of an Ascoli space which is not a $k_R$-space, namely $C_p(ω_1)$. We show that $C_k(κ)$ is Ascoli iff $cf(κ)$ is countable iff $C_k(κ)$ is metrizable.

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