Weight of precompact subsets and tightness

We prove that if K is a compact space and the space P(K K) of regular probability measures on K K has countable tightness in its weak topology, then L1( ) is separable for every 2 P(K). It has been known that such a result is a consequence of Martin's axiom MA(!1). Our theorem has several consequences; in particular, it general- izes a theorem due to Bourgain and Todor£evi¢ on measures on Rosenthal compacta. 1. Introduction. The tightness of a topological space X, denoted by (X), is the least cardinal number such that for every A X and x2 A there is a set A0 A withjA0j (X) and such that x2 A0. Throughout, K stands for a compact Hausdor topological space. By

Spaces which are generated by discrete sets

Cascales, Kakol, Saxon proved [3] that in a large class G of locally convex spaces (lcs) (containing (LM)-spaces and (DF)-spaces) for a lcs E 2 G the weak topology s(E, E') of E has countable tightness iff its weak dual (E', s(E', E)) is K-analytic. Applying examples of Pol [9] (and Kunen [12]) one gets that there exist Banach spaces C(X) over a compact scattered space X such that C(X) is not weakly K-analytic (even not weakly K-countably determined under (CH)) but the weak dual of C(X) has countable tightness. This provides also an example showing that (gDF)-spaces need not be in class G.

Let L be either the field of real numbers or a nonarchimedean rank-one valued field. For topological space T we study the Gelfand subalgebra CO(T, L) of the algebra of all L-valued continuous functions C(T, L). The main result is that if T is a paracompact locally compact Hausdorff space, which is ultraregular if L is nonarchimedean, then for f E C(T, L) the following statements are equivalent: (1) There exists a compact set K C T such that f(T K) is finite, (2) f has finite range on every discrete closed subset of T, and (3) f E CO(T, L). Throughout this paper L will stand for either the valued field R or a nonarchimedean rank-one valued field, and T for a Hausdorff topological space. T will be assumed completely regular in the real case and ultraregular in the nonarchimedean case. We will denote by C(T, L), or simply by C if there is no confusion, the algebra over L consisting of all L-valued continuous functions on T, and by CK the ideal of those continuous f E C with compact support. Let 9DT be the set of maximal ideals of C, and denote by C0 the Gelfand subalgebra of C, consisting of all f e C with the property that, for each M E 'DT, there exists X E L such that (f X) E M. (The concept of Gelfand subalgebras of general algebras has been introduced by N. Shell -formerly N. Shilkret-in [6].) We will denote by CF the subalgebra of C consisting of those f E C for which there exists a compact set K C T such that f(T K) is finite. PROPOSITION 1. For any f E CO, f(T) is compact. PROOF. First, we prove that f(T) is a precompact set. Real case. It suffices to see that f is bounded, and this follows from [5, 5.7(b)]. Nonarchimedean case. Take > 0. Note that any two (closed-open) --radius spheres B(a) = ( LI IE A a I < e} are equal or disjoint. Choose an indexed set (aci),I inf(T) such that B(aci),EI is disjoint and covers f(T). We claim I is finite. In fact, assume, to the contrary, that I is infinite. Let Ai = U,#,f'-(B(a,)). Since the family of closed-open sets (A,)iEI has the finite intersection property, there exists M E 91T such that Ai E Z[M] for any i E I. On the other hand, since f E C0 there exists X E L such that (fX) C M and hence Z(fx) n A, # 0 for any i E I, which is a contradiction. Thus I is finite and so f(T) is precompact. Received by the editors April 20, 1982 and, in revised form, February 1, 1983. 1980 Mathematics Subject Classification. Primary 54C40.

In metrizable spaces, points in the closure of a subsetAAare limits of sequences inAA; i.e., metrizable spaces are Fréchet-Uryshon spaces. The aim of this paper is to prove that metrizability and the Fréchet-Uryshon property are actually equivalent for a large class of locally convex spaces that includes(LF)(LF)- and(DF)(DF)-spaces. We introduce and study countable bounded tightness of a topological space, a property which implies countable tightness and is strictly weaker than the Fréchet-Urysohn property. We provide applications of our results to, for instance, the space of distributionsD′(Ω)\mathfrak {D}’(\Omega ). The spaceD′(Ω)\mathfrak {D}’(\Omega )is not Fréchet-Urysohn, has countable tightness, but its bounded tightness is uncountable. The results properly extend previous work in this direction.

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper first study some properties of weak topology on modular space and Convergent in weak topology on a modular space and prove some theory.

Let E be a Hausdorff topological vector space having a Schauder basis $\{ {b_i}\}$ and coordinate functionals $\{ {f_i}\}$. Let $\sigma (E,F)$ be the weak topology on E induced by $F = \{ {f_i}:i \in {\mathbf {N}}\}$. We show that if a series in E is subseries convergent with respect to $\sigma (E,F)$, then it is subseries convergent with respect to the original topology of E.

Let S be a nonconvex weakly compact and weakly connected subset of a real locally convex topological linear space L and D a relatively weakly open subset of S containing the set Inc of local nonconvexity points of S with respect to the weak topology. It is proved that ker , where regS denotes the set of regular points of S and . This substantially stregthens a recent result of Stavrakas in which the intersection above was taken over the whole set regS. The intersection formula is shown to hold also for a nonconvex connected weakly compact subset S of L with D being a relatively weakly open subset of S containing the set IncS of local nonconvexity points of S.

Let E be a separated locally convex barrelled space with continuous dual E′ and algebraic dual E* and let M be a subspace of E* with and dim Robertson, Tweddle and Yeomans have recently considered the question of barrelledness under the Mackey topology τ(E,E' + M) when E is given to be barrelled under its original topology τ(E,E') [5], [6], [7].

The Fourier binest algebra is defined as the intersection of the Volterra nest algebra on L2(ℝ) with its conjugate by the Fourier transform. Despite the absence of nonzero finite rank operators this algebra is equal to the closure in the weak operator topology of the Hilbert–Schmidt bianalytic pseudo-differential operators. The (non-distributive) invariant subspace lattice is determined as an augmentation of the Volterra and analytic nests (the Fourier binest) by a continuum of nests associated with the unimodular functions exp(−isx2/2) for s>0. This multinest is the reflexive closure of the Fourier binest and, as a topological space with the weak operator topology, it is shown to be homeomorphic to the unit disc. Using this identification the unitary automorphism group of the algebra is determined as the semi-direct product ℝ2×κℝ for the action κt(λ, μ) =(etλ, e−t μ).

In this paper, we introduce some new properties of a topological space which are respectively generalizations of -Urysohn property. We show that countably AP property is a sufficient condition for a space being countable tightness, sequential, weakly first countable and symmetrizable, to be ACP, , first countable and semimetrizable, respectively. We also prove that countable compactness is a sufficient condition for a countably AP space to be countably . We then show that a countably compact space satisfying one of the properties mentioned here is sequentially compact. And we show that a countably compact and countably AP space is maximal countably compact if and only if it is . We finally obtain a sufficient condition for the ACP closure operator to be a Kuratowski topological closure operator and related results.

Representation of set-valued mappings

0.1. This paper presents an investigation of the following problem. Exhibit a class X of topological spaces which contains all peano spaces and which has the following properties: (1) a theory exists in each space of the class, (2) the abstract set consisting of all of any space X of the class can be topologized so as to be a member of the class X, and (3) the hyperspace thus obtained is acyclic. Since the class TP of all peano spaces does not satisfy the condition (2), it is clear that any solution of the problem lies in a generalization of peano spaces. (For theory, see Whyburn [5 ] (1), and [6 ], or Kuratowski and Whyburn [4].) One such generalization has been proposed by R. L. Moore [5]; another by one of the authors (Youngs [9 ]). Moore employed two primitive concepts: region and contiguity (compare this with satelliticity 4.8). Youngs used the notion of arc as primitive. In the following pages a solution is given which is based upon the usual concept of open set. 0.2. The work is divided into four sections. The first of these is devoted to the definition and a brief discussion of the class X of spaces to be used in the remainder of the paper; namely, locally connected topological spaces. No separation or countability properties are assumed. Thus, in particular, a single need not form a closed set. The use of such a weak topology is not dictated merely by a desire for generality; indeed, it is shown in later sections that this weakness is fundamental in the consideration of hyperspaces of peano spaces. The development of a theory of elements for spaces of the class X occupies the second section. In such spaces, the standard definitions of cut point and cyclic element (Kuratowski and Whyburn [4]) fail to yield certain important properties of these concepts. However, the properties are easily recovered by generalizations of the definitions mentioned. The degree of similarity achieved between the structure of spaces of the general class X and of peano spaces seems remarkable in view of the weak topology assumed in the former.

On continuous image averaging of Borel measures