Infinite Compact Subsets Research Articles

On subspaces of spaces C_p(X) isomorphic to spaces c_{0} and ell _{q} with the topology induced from mathbb {R}^{mathbb {N}}

The linear space of all continuous real-valued functions on a Tychonoff space X with the pointwise topology (induced from the product space mathbb {R}^X) is denoted by C_p(X). In this paper we continue the systematic study of sequences spaces c_{0} and ell _{q} (for 0<qle infty ) with the topology induced from mathbb {R}^{mathbb {N}} (denoted by (c_{0})_p and (ell _{q})_{p}, respectively) and their role in the theory of C_p(X) spaces. For every infinite Tychonoff space X we construct a subspace F of C_p(X) that is isomorphic to (c_{0})_p; if X contains an infinite compact subset, then the copy F of (c_{0})_p is closed in C_p(X). It follows that C_p(X) contains a copy of (ell _{q})_{p} for every 0<qle infty . We prove that for any infinite compact space X the space C_p(X) contains no closed copy of (ell _{q})_{p} for qin (0, 1]cup {infty } and no complemented copy for 0<qle infty . Relation with results of Talagrand, Haydon, Levy and Odell will be also discussed. Examples and open problems will be provided.

Is the Free Locally Convex Space L(X) Nuclear?

Given a class $${{\mathcal {P}}}$$ of Banach spaces, a locally convex space (LCS) E is called 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 Außenhofer (Topol Appl 134:90–102, 2007), 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 $$k_\omega $$ -space, then L(X) is a Schwartz LCS (Außenhofer et al. in Stud Math 181(3):199–210, 2007), hence L(X) is multi-reflexive. We show that for every first-countable paracompact (in particular, for every metrizable) space X the converse is true, so L(X) is multi-reflexive if and only if X is a $$k_\omega $$ -space, equivalently if X is a locally compact and $$\sigma $$ -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 $$\sigma $$ -compact.

On metrizable subspaces and quotients of non-Archimedean spaces C_p(X, {mathbb {K}})

Let {mathbb {K}} be a non-trivially valued non-Archimedean complete field. Let ell _{infty }({mathbb {N}}, {mathbb {K}}) [ell _c({mathbb {N}}, {mathbb {K}});c_0({mathbb {N}}, {mathbb {K}})] be the space of all sequences in {mathbb {K}} that are bounded [relatively compact; convergent to 0] with the topology of pointwise convergence (i.e. with the topology induced from {mathbb {K}}^{{mathbb {N}}}). Let X be an infinite ultraregular space and let C_p(X,{mathbb {K}}) be the space of all continuous functions from X to {mathbb {K}} endowed with the topology of pointwise convergence. It is easy to see that C_p(X,{mathbb {K}}) is metrizable if and only if X is countable. We show that for any X [with an infinite compact subset] the space C_p(X,{mathbb {K}}) has an infinite-dimensional [closed] metrizable subspace isomorphic to c_0({mathbb {N}}, {mathbb {K}}). Next we prove that C_p(X,{mathbb {K}}) has a quotient isomorphic to c_0({mathbb {N}}, {mathbb {K}}) if and only if it has a complemented subspace isomorphic to c_0({mathbb {N}}, {mathbb {K}}). It follows that for any extremally disconnected compact space X the space C_p(X,{mathbb {K}}) has no quotient isomorphic to the space c_0({mathbb {N}}, {mathbb {K}}); in particular, for any infinite discrete space D the space C_p(beta D, {mathbb {K}}) has no quotient isomorphic c_0({mathbb {N}}, {mathbb {K}}). Finally we investigate the question for which X the space C_p(X,{mathbb {K}}) has an infinite-dimensional metrizable quotient. We show that for any infinite discrete space D the space C_p(beta D, {mathbb {K}}) has an infinite-dimensional metrizable quotient isomorphic to some subspace ell _c^0({mathbb {N}}, {mathbb {K}}) of {mathbb {K}}^{{mathbb {N}}}. If {mathbb {K}} is locally compact then ell _c^0({mathbb {N}}, {mathbb {K}})=ell _{infty }({mathbb {N}}, {mathbb {K}}). If |n1_{{mathbb {K}}}|ne 1 for some nin {mathbb {N}}, then ell _c^0({mathbb {N}}, {mathbb {K}})=ell _c ({mathbb {N}}, {mathbb {K}}). In particular, C_p(beta D, {mathbb {Q}}_q) has a quotient isomorphic to ell _{infty }({mathbb {N}}, {mathbb {Q}}_q) and C_p(beta D, {mathbb {C}}_q) has a quotient isomorphic to ell _c({mathbb {N}}, {mathbb {C}}_q) for any prime number q.

On convergent sequences in dual groups

We provide some characterizations of precompact abelian groups $G$ whose dual group $G_p^\wedge$ endowed with the pointwise convergence topology on elements of $G$ contains a nontrivial convergent sequence. In the special case of precompact abelian \emph{torsion} groups $G$, we characterize the existence of a nontrivial convergent sequence in $G_p^\wedge$ by the following property of $G$: \emph{No infinite quotient group of $G$ is countable.} Finally, we present an example of a dense subgroup $G$ of the compact metrizable group $\mathbb{Z}(2)^\omega$ such that $G$ is of the first category in itself, has measure zero, but the dual group $G_p^\wedge$ does not contain infinite compact subsets. This complements Theorem 1.6 in [J.E.~Hart and K.~Kunen, Limits in function spaces and compact groups, \textit{Topol. Appl.} \textbf{151} (2005), 157--168]. As a consequence, we obtain an example of a precompact reflexive abelian group which is of the first Baire category.

Point derivations and cohomologies of Lipschitz algebras

AbstractFor a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.

Null-finite sets in topological groups and their applications

In the paper we introduce a new family of "small" sets which is tightly connected with two well known $\sigma$-ideals: of Haar-null sets and of Haar-meager sets. We define a subset $A$ of a topological group $X$ to be $\mathit{null}$-$\mathit{finite}$ if there exists an infinite compact subset $K\subset X$ such that for every $x\in X$ the intersection $K\cap (x+A)$ is finite. We prove that each null-finite Borel set in a complete metric Abelian group is Haar-null and Haar-meager. The Borel restriction in the above result is essential as each non-discrete metric Abelian group is the union of two null-finite sets. Applying null-finite sets to the theory of functional equations and inequalities, we prove that a mid-point convex function $f:G\to\mathbb R$ defined on an open convex subset $G$ of a metric linear space $X$ is continuous if it is upper bounded on a subset $B$ which is not null-finite and whose closure is contained in $G$. This gives an alternative short proof of a known generalization of Bernstein-Doetsch theorem (saying that a mid-point convex function $f:G\to\mathbb R$ defined on an open covex subset $G$ of a metric linear space $X$ is continuous if it is upper bounded on a non-empty open subset $B$ of $G$). Since Borel null-finite sets are Haar-meager and Haar-null, we conclude that a mid-point convex function $f:G\to\mathbb{R}$ defined on an open convex subset $G$ of a complete linear metric space $X$ is continuous if it is upper bounded on a Borel subset $B\subset G$ which is not Haar-null or not Haar-meager in $X$. The last result resolves an old problem in the theory of functional equations and inequalities posed by Baron and Ger in 1983.

Metrizable quotients of Cp-spaces

The famous Rosenthal–Lacey theorem asserts that for each infinite compact set K the Banach space C(K) admits a quotient which is either an isomorphic copy of c or ℓ2. What is the case when the uniform topology of C(K) is replaced by the pointwise topology? Is it true that Cp(X) always has an infinite-dimensional separable (or better metrizable) quotient? In this paper we prove that for a Tychonoff space X the function space Cp(X) has an infinite-dimensional metrizable quotient if X either contains an infinite discrete C⁎-embedded subspace or else X has a sequence (Kn)n∈N of infinite compact subsets such that for every n the space Kn contains two disjoint topological copies of Kn+1. Applying the latter result, we show that under ◊ there exists a zero-dimensional Efimov space K whose function space Cp(K) has an infinite-dimensional metrizable quotient. These two theorems essentially improve earlier results of Ka̧kol and Śliwa on infinite-dimensional separable quotients of Cp-spaces.

A characterisation of the Daugavet property in spaces of Lipschitz functions

We study the Daugavet property in the space of Lipschitz functions Lip0(M) on a complete metric space M. Namely we show that Lip0(M) has the Daugavet property if and only if M is a length metric space. This condition also characterises the Daugavet property in the Lipschitz free space F(M). Moreover, when M is compact, we show that either F(M) has the Daugavet property or its unit ball has a strongly exposed point. If M is an infinite compact subset of a strictly convex Banach space then the Daugavet property of Lip0(M) is equivalent to the convexity of M.

A countable free closed non-reflexive subgroup of ℤ^{𝔠}

We prove that the group G = H o m ( Z N , Z ) G=\mathrm {Hom}(\mathbb {Z}^{\mathbb {N}}, \mathbb {Z}) of all homomorphisms from the Baer-Specker group Z N \mathbb {Z}^{\mathbb {N}} to the group Z \mathbb {Z} of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G G is discrete. As G G is non-discrete, it is not reflexive. Since G G can be viewed as a closed subgroup of the Tychonoff product Z c \mathbb {Z}^{\mathfrak {c}} of continuum many copies of the integers Z \mathbb {Z} , this provides an example of a group described in the title, thereby resolving a problem by Galindo, Recoder-Núñez and Tkachenko. It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.

Domination in products

In this note we continue to study the cardinal invariants dm and sdm introduced by the author in [7]. We prove that if K is a compact subspace of Cp(Y) for some space Y such that dm(Y)≤κ then K is strongly κ-monolithic. Also we show how GCH implies that if sdm(Cp(X))≤ω for some hereditarily normal space X of character at most c then every infinite compact subset of X is countable. Finally we show that for every cardinal κ there is a metric space of weight at most κ that condenses onto Σ(2κ).

Reflexivity in precompact groups and extensions

We establish some general principles and find some counter-examples concerning the Pontryagin reflexivity of precompact groups and P-groups. We prove in particular that:(1)A precompact Abelian group G of bounded order is reflexive iff the dual group G∧ has no infinite compact subsets and every compact subset of G is contained in a compact subgroup of G.(2)Any extension of a reflexive P-group by another reflexive P-group is again reflexive. We show on the other hand that an extension of a compact group by a reflexive ω-bounded group (even dual to a reflexive P-group) can fail to be reflexive.We also show that the P-modification of a reflexive σ-compact group can be non-reflexive (even if, as proved in [20], the P-modification of a locally compact Abelian group is always reflexive).

Ratio Asymptotics, Hessenberg Matrices, and Weak Asymptotic Measures

We discuss the relationship between ratio asymptotics for general orthogonal polynomials and the asymptotics of the associated Bergman shift operator. More specifically, we consider the case in which a measure is supported on an infinite compact subset of the complex plane. We show that there is a straightforward connection between the corresponding orthonormal polynomials exhibiting ratio asymptotics and the corresponding Bergman shift operator being asymptotically Toeplitz. We also discuss a connection to the weak asymptotics of the measures derived from the orthonormal polynomials.

Asymptotics of Discrete Riesz d-Polarization on Subsets of d-Dimensional Manifolds

We prove a conjecture of T. Erdélyi and E.B. Saff, concerning the form of the dominant term (as N → ∞) of the N-point Riesz d-polarization constant for an infinite compact subset A of a d-dimensional C 1-manifold embedded in ℝ m (d ≤ m). Moreover, if we assume further that the d-dimensional Hausdorff measure of A is positive, we show that any asymptotically optimal sequence of N-point configurations for the N-point d-polarization problem on A is asymptotically uniformly distributed with respect to $\mathcal H_d|_A$ . These results also hold for finite unions of such sets A provided that their pairwise intersections have $\mathcal H_d$ -measure zero.

More on convergent sequences in free topological groups

We show that, for every infinite cardinal τ, there exists a zero-dimensional pseudocompact space X with |X|=τω such that all countable subsets of X are closed (hence all compact subsets of X are finite), but both the free Abelian topological group A(X) and the free topological group F(X) on X contain a compact subspace of cardinality τ. This answers a recent question raised by A.V. Arhangelʼskii. We also show that there exists a countably compact space Y without infinite compact subsets such that the groups F(Y) and A(Y) contain the one-point compactification of a discrete space of cardinality 2c (under additional set-theoretic assumptions, the groups F(Y) and A(Y) can contain arbitrarily big compact subsets).If, however, X is homeomorphic to a topological group, then the existence of an infinite compact subset of F(X) (or A(X)) implies that X contains an infinite compact subset as well.

Topologies as points within a Stone space: Lattice theory meets topology

For a non-empty set X, the collection Top(X) of all topologies on X sits inside the Boolean lattice P(P(X)) (when ordered by set-theoretic inclusion) which in turn can be naturally identified with the Stone space 2P(X). Via this identification then, Top(X) naturally inherits the subspace topology from 2P(X). Extending ideas of Frink (1942), we apply lattice-theoretic methods to establish an equivalence between the topological closures of sublattices of 2P(X) and their (completely distributive) completions. We exploit this equivalence when searching for countably infinite compact subsets within Top(X) and in crystallizing the Borel complexity of Top(X). We exhibit infinite compact subsets of Top(X) including, in particular, copies of the Stone–Čech and one-point compactifications of discrete spaces.

Pontryagin duality in the class of precompact Abelian groups and the Baire property

We present a wide class of reflexive, precompact, non-compact, Abelian topological groups G determined by three requirements. They must have the Baire property, satisfy the open refinement condition, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group G∧ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compact Abelian group G of weight ≥2ω, we find proper dense subgroups H1 and H2 of G such that H1 is reflexive and pseudocompact, while H2 is non-reflexive and almost metrizable.

Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

Let (X, d) be a pointed compact metric space, let 0 &lt; α &lt; 1, and let φ : X → X be a base point preserving Lipschitz map. We prove that the essential norm of the composition operator Cφ induced by the symbol φ on the spaces lip0(X, dα) and Lip0(X, dα) is given by the formula whenever the dual space has the approximation property. This happens in particular when X is an infinite compact subset of a finite‐dimensional normed linear space.

Pseudocompact group topologies with no infinite compact subsets

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property ♯ ). Every pseudocompact Abelian group G with cardinality | G | ≤ 2 2 c satisfies this inequality and therefore admits a pseudocompact group topology with property ♯ . Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property ♯ . We also observe that pseudocompact Abelian groups with property ♯ contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.

Forcing hereditarily separable compact-like group topologies on Abelian groups

Let c denote the cardinality of the continuum. Using forcing we produce a model of ZFC + CH with 2 c “arbitrarily large” and, in this model, obtain a characterization of the Abelian groups G (necessarily of size at most 2 c ) which admit: (i) a hereditarily separable group topology, (ii) a group topology making G into an S-space, (iii) a hereditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no infinite compact subsets), (iv) a group topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without infinite compact subsets if necessary). As a by-product, we completely describe the algebraic structure of the Abelian groups of size at most 2 c which possess, at least consistently, a countably compact group topology (without infinite compact subsets, if desired). We also put to rest a 1980 problem of van Douwen about the cofinality of the size of countably compact Abelian groups.