Countable Subfamily Research Articles

On $\mathbb R$-embeddability of almost disjoint families and Akemann–Doner C$^*$-algebras

An almost disjoint family $\mathcal A$ of subsets of $\mathbb N$ is said to be $\mathbb R$-embeddable if there is a function $f:\mathbb N\rightarrow \mathbb R$ such that the sets $f[A]$ are ranges of real sequences converging to distinct reals for distinct $A\in \mathcal A$. It is well known that almost disjoint families which have few separations, such as Luzin families, are not $\mathbb R$-embeddable. We study extraction principles related to $\mathbb R$-embeddability and separation properties of almost disjoint families of $\mathbb N$ as well as their limitations. An extraction principle whose consistency is our main result is: every almost disjoint family of size continuum contains an $\mathbb R$-embeddable subfamily of size continuum. It is true in the Sacks model. The Cohen model serves to show that the above principle does not follow from the fact that every almost disjoint family of size continuum has two separated subfamilies of size continuum. We also construct in ZFC an almost disjoint family, where no two uncountable subfamilies can be separated but always a countable subfamily can be separated from any disjoint subfamily. Using a refinement of the $\mathbb R$-embeddability property called a controlled $\mathbb R$-embedding property we obtain the following results concerning Akemann-Doner C*-algebras which are induced by uncountable almost disjoint families: a) In ZFC there are Akemann-Doner C*-algebras of density $\mathfrak c$ with no commutative subalgebras of density $\mathfrak c$, b) It is independent from ZFC whether there is an Akemann-Doner algebra of density $\mathfrak c$ with no nonseparable commutative subalgebra. This completes an earlier result that there is in ZFC an Akemann-Doner algebra of density $\omega_1$ with no nonseparable commutative subalgebra.

On resonances and bound states of Smilansky Hamiltonian

We consider the self-adjoint Smilansky Hamiltonian $\mathsf{H}_\varepsilon$ in $L^2(\mathbb{R}^2)$ associated with the formal differential expression $-\partial_x^2 - \frac12\big(\partial_y^2 + y^2) - \sqrt{2}\varepsilon y \delta(x)$ in the sub-critical regime, $\varepsilon \in (0,1)$. We demonstrate the existence of resonances for $\mathsf{H}_\varepsilon$ on a countable subfamily of sheets of the underlying Riemann surface whose distance from the physical sheet is finite. On such sheets, we find resonance free regions and characterise resonances for small $\varepsilon > 0$. In addition, we refine the previously known results on the bound states of $\mathsf{H}_\varepsilon$ in the weak coupling regime ($\varepsilon\rightarrow 0+$). In the proofs we use Birman-Schwinger principle for $\mathsf{H}_\varepsilon$, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.

Lattices of homomorphisms and pro-Lie groups

Early this century K. H. Hofmann and S. A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, all locally compact abelian groups, and all connected locally compact groups and is closed under the formation of products and closed subgroups. They defined a topological group G to be almost connected if the quotient group of G by the connected component of its identity is compact.We show here that all almost connected pro-Lie groups as well as their continuous homomorphic images are R-factorizable and ω-cellular, i.e. every family of Gδ-sets contains a countable subfamily whose union is dense in the union of the whole family. We also prove a general result which implies as a special case that if a topological group G contains a compact invariant subgroup K such that the quotient group G/K is an almost connected pro-Lie group, then G is R-factorizable and ω-cellular.Applying the aforementioned result we show that the sequential closure and the closure of an arbitrary Gδ,Σ-set in an almost connected pro-Lie group H coincide.

Decomposition and construction of fuzzy sets and their applications to the arithmetic operations on fuzzy quantities

The membership function of a fuzzy set can be constructed from a family of subsets of a universal set via the form of resolution identity assuming that the family is nested. In this paper, we are going to consider the weak case of assuming that only a countable subfamily is nested. Using this generalized consideration, we can overcome the ambiguity of arithmetic operations on fuzzy quantities. Two well-known additions of fuzzy quantities in vector space have been adopted in the literature. One is based on the extension principle by directly considering the membership functions. Of course, this addition always exists. Another one uses the α-level sets without considering the membership functions. However, the existence of this addition cannot be guaranteed. In this paper, we are going to propose two other additions of fuzzy quantities in vector space. One generalizes the extension principle, and the other one also uses the α-level sets without considering the membership functions such that this generalized addition always exists. We can show that these four additions are equivalent.

Helly-type theorems for intersections of sets starshaped via orthogonally convex paths

Let \({\mathcal{K}}\) be a family of simply connected sets in the plane. If every countable subfamily of \({\mathcal{K}}\) has an intersection that is starshaped via orthogonally convex paths, then \({\mathcal{K}}\) itself has such an intersection. For the d-dimensional case, let \({\mathcal{K}}\) be a family of compact sets in \({\mathbb{R}^d}\). If every finite subfamily of \({\mathcal{K}}\) has an intersection that is starshaped via orthogonally convex paths, again \({\mathcal{K}}\) itself has such an intersection.

Lindelöf type of generalization of separability in Banach spaces

We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: X has CSP if each family E of closed linear subspaces of X whose intersection is the zero space contains a countable subfamily E 0 with the same intersection. All separable Banach spaces have CSP and plenty of examples of non-separable CSP spaces are provided. Connections of CSP with Markučevič-bases, Corson property and related geometric issues are discussed.

Algebraic Characterization of Dense-Separability Among Compact Spaces

In this note, we show that a compact Hausdorff space X is dense-separable if and only if every family of ideals of C(X) with zero intersection has a countable subfamily with zero intersection. As a consequence of this characterization we observe that every compact dense-separable space whith Soc(C(X)) = 0 has a countable dense and co-dense subset.

On I-Lindelöf sets

A space (X, T) is called I-Lindelof [1] if every cover A of X by regular closed subsets of the space (X, T) contains a countable subfamily A′ such that X = ∪{int (A): A ∈ A′}. In this work we introduce the class of I-Lindelof sets as a proper subclass of rc-Lindelof sets [3]. We study various properties of I-Lindelof sets and investigate the relationship between I-Lindelof sets and I-Lindelof subspaces. We give a new characterization of I-Lindelof spaces in terms of this type of sets. Also, we study spaces (X, T) in which every I-Lindelof set in (X, T) is closed.

κ-Fréchet Urysohn property of [formula omitted

For a Tychonoff space X, we denote by C k ( X ) the space of all real-valued continuous functions on X with the compact open topology. A space X is said to be κ-Fréchet Urysohn if for every open subset U of X and every x ∈ U ¯ , there exists a sequence { x n } n ∈ ω ⊂ U converging to x. In this paper, we show that C k ( X ) is κ-Fréchet Urysohn iff every moving off family of compact subsets of X has a countable subfamily which is strongly compact-finite. In particular, we obtain that every stratifiable Baire space C k ( X ) is an M 1 -space.

On relatively weakly almost Lindelöf subsets

Summary A subset Y of a space X is weakly almost Lindelf in X if for every open cover U of X, there exists a countable subfamily V of U such that Y ⊆ comp (∩V ). We investigate the relationship between relatively weakly almost Lindelf subsets and relatively almost Lindelf subsets, and also study various properties of relatively weakly almost Lindelf subsets.

A Helly-type theorem for countable intersections of starshaped sets

Let k and d be fixed integers, 0≦k≦d, and let $\mathcal{K} = \{ k_\alpha :\alpha {\text{ in some index set}}\} $ be a collection of sets in $\mathbb{R}^d .$ If every countable subfamily of $\mathcal{K}$ has a starshaped intersection, then $ \cap \{ k_\alpha :k_\alpha {\text{ in }}\mathcal{K}\} $ is (nonempty and) starshaped as well. Moreover, if every countable subfamily of $\mathcal{K}$ has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of $ \cap \{ k_\alpha :k_\alpha {\text{ in }}\mathcal{K}\} $ is at least k-dimensional, too. Finally, dual statements hold for unions of sets.

On first and second countable spaces and the axiom of choice

In this paper it is studied the role of the axiom of choice in some theorems in which the concepts of first and second countability are used. Results such as the following are established: (1) In ZF (Zermelo–Fraenkel set theory without the axiom of choice), equivalent are: (i) every base of a second countable space has a countable subfamily which is a base; (ii) the axiom of countable choice for sets of real numbers. (2) In ZF, equivalent are: (i) every local base at a point x, in a first countable space, contains a countable base at x; (ii) the axiom of countable choice ( CC). (3) In ZF, equivalent are: (i) for every local base system ( B(x)) x∈X of a first countable space X, there is a local base system ( V(x)) x∈X such that, for each x∈ X, V(x) is countable and V(x)⊆ B(x) ; (ii) for every family ( X i ) i∈ I of non-empty sets there is a family ( A i ) i∈ I of non-empty, at most countable sets, such that A i ⊆ X i for every i∈ I ( ω- MC) and CC.

I‐Lindelof spaces

We define a space (X, T) to be I‐Lindelof if every cover 𝒜 of X by regular closedsubsetsof the space (X, T) contains a countable subfamily 𝒜′ suchthat X = ∪{int(A) : A ∈ A′}. We provide several characterizations of I‐Lindelofspaces and relate them to some other previously known classes of spaces, for example, rc‐Lindelof, nearly Lindelof, and so forth. Our study here of I‐Lindelofspaces also deals with operations on I‐Lindelof spaces and, in its last part, investigates images and inverseimagesof I‐Lindelofspacesundersome consideredtypesof functions.

Omitting types in incomplete theories

AbstractWe characterize omissibility of a type, or a family of types, in a countable theory in terms of non-existence of a certain tree of formulas. We extend results of L. Newelski on omitting < covK non-isolated types. As a consequence we prove that omissibility of a family of < covK types is equivalent to omissibility of each countable subfamily.

𝐹_{𝜎}-set covers of analytic spaces and first class selectors

Let X X be an analytic space (e.g., a complete metric space). We prove that any point-countable F σ {F_\sigma } -set cover of X X either has σ \sigma -discrete refinement, or else there is a compact subset of X X not covered by any countable subfamily of the cover. It follows that any point-countable F σ {F_\sigma } -additive family in X X has a σ \sigma -discrete refinement. This is used to show that any weakly F σ {F_\sigma } -measurable multimap, defined on X X and taking nonempty, closed and separable values in a complete metric space, has a selector of the first Baire class.

Generalized products of weakly m — n compact spaces, I

AbstractA topological space X is said to be weakly-Lindelöf if and only if every open cover of X has a countable sub-family with dense union. We know that products of two Lindelöf spaces need not be weakly-Lindelöf. In this paper we obtain non-trivial sufficient conditions on small sub-products to ensure the producitivity of the property weakly-Lindelöf with respect to arbitrary products.

Errata to ‘‘General product measures”

Theorem 5.9 is false and the proof we gave of Theorem 7.7 used 5.9. Fortunately a proof of 7.7, known earlier to us, makes no use of 5.9. For the record we also note that the conclusion about B in Theorem 7.2.4 is also false but this was not used anywhere in the paper. The proof of Theorem 7.7 can be repaired by replacing Part 2 thereof (p. 282) by the following. PART 2. If a-c and ag = S then there is such a countable subfamily ( of a that . V(S, a(M) = 0. Proof. The desired conclusion is found in Step 2 below. Let V =tpr u whenever u e fnt r sb t and Fp=EA (for some Be and uct, dmn u = dmn p, A E VU, and cyl AS' B) whenever p E fnt n sb m. Now we take STEP 1. If p E fnt n sb m then there is such a countable subfamily ( of a that

Some Problems in the Theory of Optimal Stopping Rules

Let $y_1, y_2, \cdots$ be a sequence of random variables with known joint distribution. We are allowed to observe the $y$'s sequentially. We must terminate the observation process at some point, and if we stop at time $n$, we receive a reward which is a known function of $y_1, \cdots, y_n$. Our decision to stop at time $n$ is allowed to depend on the observations we have previously made but may not depend on the future, which is still unknown. We are interested in finding stopping rules which maximize our expected terminal reward. More formally, let $(x_n, F_n)_{1 \leqq n}$ be a stochastic sequence on a probability space $(W, F, P)$, i.e., let $(F_n)$ be an increasing sequence of sub-sigma-algebras of $F$ and for each $n \leqq 1$ let $x_n$ be a random variable (rv) measurable $F_n$. In terms of the intuitive background of the preceding paragraph, $F_n = B(y_1, \cdots, y_n)$; and although it is convenient to keep this interpretation in mind, our general results do not depend on it. A stopping rule or stopping variable (sv) is a rv $t$ with values $1, 2, \cdots, +\infty$, such that $P(t < \infty) = 1$ and for each $n \geqq 1 (t = n) \varepsilon F_n. x_t$ is (up to an equivalence) a rv, and if $v = \sup Ex_t$, where the supremum is taken over all sv's such that $Ex_t$ exists, we are interested in answering the following questions: (a) What is $v$? (b) Is there an optimal sv, i.e., one for which $Ex_t$ exists and equals $v$? (c) If there exists an optimal sv, what is it? The problem stated above is not sufficiently well formulated, as the class of sv's $t$ such that $Ex_t$ exists may be vacuous. To avoid this and other uninteresting complications we shall add the assumption that $E|x_n| < \infty, n \geqq 1$. We recall that the essential supremum (e. sup) of a family of rv's $\{q_t, t \varepsilon T\}$ is a rv $Q$ such that (1) $Q \geqq q_t$ a.s. $t \varepsilon T$, and (2) if $Q'$ is any rv such that $Q' \geqq q_t$ a.s., $t \varepsilon T$, then $Q' \geqq Q$ a.s. It is known that the essential supremum of a family of rv's always exists and can be assumed to be the supremum of some countable subfamily (e.g., [12], p. 44). Let $C_n$ be the class of all sv's $t$ such that $P(t \geqq n) = 1$ and $EX^-_t < \infty$. Let $f_n = e. \sup_{t \varepsilon C_n} E(x_t \mid F_n), v_n = \sup_{C_n} Ex_t$. It is known (Theorem 2 of [3]) that if $v < \infty$ and an optimal rule exists, then $s = \text{first} n \geqq 1$ such that $x_n = f_n$ is an optimal rule. For this and various other reasons which will become apparent, e.g., Theorem 1 below, it is desirable to have a constructive method for computing the $f_n$. The technique of backward induction and taking limits, originating with [1] and described in Theorem 2 below, achieves the desired result under certain conditions (see Theorem 2 of [4] for a general statement of these conditions). The central theorem of Section 2 provides completely general methods for computing the $f_n$. Although it seems unlikely that one would ever find it desirable to carry out these computations, there are, nevertheless, several interesting applications of the results to the theory of optimal stopping rules, and it is these applications which concern us throughout the remainder of this paper. In the course of these investigations we find it convenient to introduce the notion of an extended sv, i.e., we drop the requirement that $t$ be finite with probability one while defining $x_\infty$ to be $\lim \sup x_n$. We show that $\bar f_n = f_n$, where, relative to the class of extended sv's, $\bar f_n$ is defined analogously to $f_n$. We utilize extended sv's as a technical device within the framework of the usual theory and give examples which illustrate the inherent value of these sv's. In Section 3 we define the Markov case. We show that by paying proper attention to the Markovian structure of many stopping rule problems we are able to simplify somewhat the general theory and to give relatively simple descriptions of optimal rules when they exist. We also define randomized sv's and show that randomization does not increase $v$. We then apply this result to prove the monotonocity and continuity of $v = v(p)$ in the case where $x_n$ is the proportion of heads in $n$ independent tosses of a coin having probability $p$ of heads on each toss.