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.
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.
Following [3] we say that a Tychonoff space $X$ is an Ascoli space if every compact subset $\mathcal{K}$ of $C_k(X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_\mathbb{R}$-space, hence any $k$-space, is Ascoli. Let $X$ be a metrizable space. We prove that the space $C_{k}(X)$ is Ascoli iff $C_{k}(X)$ is a $k_\mathbb{R}$-space iff $X$ is locally compact. Moreover, $C_{k}(X)$ endowed with the weak topology is Ascoli iff $X$ is countable and discrete. Using some basic concepts from probability theory and measure-theoretic properties of $\ell_1$,…
We characterize Ascoli spaces by showing that a Tychonoff space $X$ is Ascoli iff the canonical map from the free locally convex space $L(X)$ over $X$ into $C_k\big(C_k(X)\big)$ is an embedding of locally convex spaces. We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli. If a $c_0$-barrelled space $X$ is weakly Ascoli, then $X$ is linearly isomorphic to a dense subspace of $\mathbb{R}^Γ$ for some $Γ$. Consequently, a Fréchet space $E$ is weakly Ascoli iff $E=\mathbb{R}^N$ for some $N\leqω$. If $X$ is a $μ$-space and a $k$-space (for example, metrizable),…
The paper deals with Ascoli spaces $C_p(X)$ and $C_k(X)$ over Tychonoff spaces $X$. The class of Ascoli spaces $X$, i.e. spaces $X$ for which any compact subset $K$ of $C_k(X)$ is evenly continuous, essentially includes the class of $k_{\mathbb R}$-spaces. First we prove that if $C_p(X)$ is Ascoli, then it is $κ$-Fréchet-Urysohn. If $X$ is cosmic, then $C_p(X)$ is Ascoli iff it is $κ$-Fr'echet-Urysohn. This leads to the following extension of a result of Morishita: If for a Čech-complete space $X$ the space $C_p(X)$ is Ascoli, then $X$ is scattered. If $X$ is scattered and stratifiable, then $…
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 metri…
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.
The purpose of this paper is to generalize the measure of non-compactness for the space of continuous functions over the $T_{3 \frac{1}{2}}$ space. Motivated by the generalized Arzelà-Ascoli theorem for Tichonoff space $T$ via Wallman compactifiaction $\operatorname{Wall}(T)$, we constuct a measure of non-compactness for the space $C^b(T)$. We also study some of the properties of this object and give another version of Darbo-type theorem, suitable for this particular case.
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 distribution…
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.
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…
A topological space $X$ is $κ$-Fréchet--Urysohn if for every open subset $U$ of $X$ and every $x\in \overline{U}$ there exists a sequence in $ U$ converging to $x$. We prove that every $κ$-Fréchet--Urysohn Tychonoff space $X$ is Ascoli. We apply this statement and some of known results to characterize the $κ$-Fréchet--Urysohn property in various important classes of locally convex spaces. In particular, answering a question posed in [7] we obtain that $C_p(X)$ is Ascoli iff $X$ has the property $(κ)$.
The paper studies the free locally convex space $L(X)$ over a Tychonoff space $X$. Since for infinite $X$ the space $L(X)$ is never metrizable (even not Fréchet-Urysohn), a possible applicable generalized metric property for $L(X)$ is welcome. We propose a concept (essentially weaker than first-countability) which is known under the name a $\mathfrak{G}$-base. A space $X$ has a {\em $\mathfrak{G}$-base} if for every $x\in X$ there is a base $\{ U_α: α\in\mathbb{N}^\mathbb{N}\}$ of neighborhoods at $x$ such that $U_β\subseteq U_α$ whenever $α\leqβ$ for all $α,β\in\mathbb{N}^\mathbb{N}$, where $…
We show that the product of any number of sequentially pseudocompact topological spaces is still sequentially pseudocompact. The definition of sequential pseudocompactness can be given in (at least) two ways: we show their equivalence. Some of the results of the present note already appeared in A. Dow, J. R. Porter, R. M. Stephenson, R. G. Woods, Spaces whose pseudocompact subspaces are closed subsets, Appl. Gen. Topol. 5 (2004), 243-264.
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_{…
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 $g$-sequ…
A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel real-valued functions on Tychonoff spaces, with the topology of pointwise convergence. Our results solve a problem stated by Gartside, Lo, and Marsh.
We show that, under suitably general formulations, covering properties, accumulation properties and filter convergence are all equivalent notions. This general correspondence is exemplified in the study of products. Let $X$ be a product of topological spaces. We prove that $X$ is sequentially compact if and only if all subproducts by $\leq \mathfrak s$ factors are sequentially compact. If $\mathfrak s = \mathfrak h$, then $X$ is sequentially compact if and only if all factors are sequentially compact and all but at most $<\mathfrak s$ factors are ultraconnected. We give a topological proof of …
Inspired by a recent work of Dias and Tall, we show that a compact indestructible space is sequentially compact. We also prove that a Lindelof Hausdorff indestructible space has the finite derived set property and a compact Hausdorff indestructible space is pseudoradial.
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…
A space X is selectively sequentially pseudocompact if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in each U_n in such a way that the sequence (x_n) has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topolo…
Let $X$ be a zero-dimensional metric space and $X'$ its derived set. We prove the following assertions: (1) the space $C_k(X,2)$ is an Ascoli space iff $C_k(X,2)$ is $k_\mathbb{R}$-space iff either $X$ is locally compact or $X$ is not locally compact but $X'$ is compact, (2) $C_k(X,2)$ is a $k$-space iff either $X$ is a topological sum of a Polish locally compact space and a discrete space or $X$ is not locally compact but $X'$ is compact, (3) $C_k(X,2)$ is a sequential space iff $X$ is a Polish space and either $X$ is locally compact or $X$ is not locally compact but $X'$ is compact, (4) $C_k…
Let $κ$ be an infinite cardinal. A topological space $X$ is $κ$-bounded if the closure of any subset of cardinality $\leκ$ in $X$ is compact. We discuss the problem of embeddability of topological spaces into Hausdorff (Urysohn, regular) $κ$-bounded spaces, and present a canonical construction of such an embedding. Also we construct a (consistent) example of a sequentially compact separable regular space that cannot be embedded into a Hausdorff $ω$-bounded space.
For a Tychonoff space X, we denote by B(X) the space of all Baire functions on X with the topology of pointwise convergence. In this paper we investigate selectors for sequences of countable dense and countable sequentially dense sets of the space B(X).
For a Tychonoff space $X$, we will denote by $USC_{p}(X)$ ($B_1(X)$) a set of all real-valued upper semicontinuous functions (a set of all Baire functions of class 1) defined on $X$ endowed with the pointwise convergence topology. In this paper we describe a class of Tychonoff spaces $X$ for which the space $USC_{p}(X)$ is sequentially separable. Unexpectedly, it turns out that this class coincides with the class of spaces for which a stronger form of the sequential separability for the space $B_1(X)$ holds.
In this paper, we prove that in forcing extensions by a poset with finally property K over a model of GCH+$\square$, every compact sequentially compact space is weakly pseudoradial. We also prove the following assuming $\mathfrak{s}\leq \aleph_2$: (i) if $X$ is compact weakly pseudoradial, then $X$ is pseudoradial if and only if $X$ cannot be mapped onto $[0,1]^\mathfrak{s}$; (ii) if $X$ and $Y$ are compact pseudoradial spaces such that $X\times Y$ is weakly pseudoradial, then $X\times Y$ is pseudoradial. These results add to the wide variety of partial answers to the question by Gerlits and N…
In this paper we compare the concepts of pseudoradial spaces and the recently defined strongly pseudoradial spaces in the realm of compact spaces. We show that $\mathrm{MA}+\mathfrak{c}=ω_2$ implies that there is a compact pseudoradial space that is not strongly pseudoradial. We essentially construct a compact, sequentially compact space $X$ and a continuous function $f:X\toω_1+1$ in such a way that there is no copy of $ω_1+1$ in $X$ that maps cofinally under $f$. We also give some conditions that imply the existence of copies of $ω_1$ in spaces. In particular, $\mathrm{PFA}$ implies that comp…
We show that for arbitrary linearly ordered set $X$ any bounded family of (not necessarily, continuous) real valued functions on $X$ with bounded total variation does not contain independent sequences. We obtain generalized Helly's sequential compactness type theorems. One of the theorems asserts that for every compact metric space $(Y,d)$ the compact space $BV_r(X,Y)$ of all functions $X \to Y$ with variation $\leq r$ is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space $M_+(X,Y)$ of all order preserving maps $X \to Y$ is sequentially comp…
In this paper the weak topology on a normed space is studied from the viewpoint of infinite-dimensional topology. Besides the weak topology on a normed space $X$ (coinciding with the topology of uniform convergence on finite subsets of the dual space $X^*$), we consider the topology $c$ of uniform convergence on compact subsets of $X^*$. It is known that this topology coincides with the weak topology on bounded subsets of $X$, but unlike to the latter has much better topological properties (e.g., is stratifiable). We prove that for normed spaces $X,Y$ with separable duals the spaces $(X,weak)$…
