Search results for "Weak topology"
showing 10 items of 10 documents
Generalized fuzzy topology versus non-commutative topology
2011
The paper introduces a modification of the notions of generalized fuzzy topological space of Demirci and quantal space of Mulvey and Pelletier, suitable to explore interrelations between point-set lattice-theoretic topology and non-commutative topology developed in the framework of C^*-algebras or (more recently) of quantales. As a consequence of the new approach, a generalization of the concept of topological system of Vickers arises. Moreover, the currently dominating variable-basis topological setting in the fuzzy community, due to Rodabaugh, appears to be ''fixed-basis''.
Composite variety-based topological theories
2012
Motivated by the recent result of Rodabaugh on categorical redundancy of lattice-valued bitopology, the paper considers another viewpoint on the topic, based on the notion of composite variety-based topological theory. The new concept, apart from providing a variable-basis generalization of bitopology, incorporates the most important approaches to topology currently developed in the fuzzy community, bringing forward their categorically algebraic properties, which are cleared from point-set lattice-theoretic dependencies. Dwelling on different ways of interaction between composite topology and topology, e.g., embedding the former into the latter as a full bicoreflective subcategory, we final…
Deriving Reference Decisions
1998
To solve a statistical decision problem from a Bayesian viewpoint, the decision maker must specify a probability distribution on the parameter space, his prior distribution. In order to analyze the influence of this prior distribution on the solution of the problem, Bernardo (1981) proposed to compare the results with those that one would obtain by using that prior distribution which maximizes the useful experimental information, thus introducing the concept of reference decision. This definition is too involved for most of the problems usually found in practice. Here we analyze situations in which it is possible to simplify the definition of the reference decision, and we provide condition…
Banach spaces where convex combinations of relatively weakly open subsets of the unit ball are relatively weakly open
2018
We introduce and study Banach spaces which have property CWO, i.e., every finite convex combination of relatively weakly open subsets of their unit ball is open in the relative weak topology of the unit ball. Stability results of such spaces are established, and we introduce and discuss a geometric condition---property (co)---on a Banach space. Property (co) essentially says that the operation of taking convex combinations of elements of the unit ball is, in a sense, an open map. We show that if a finite dimensional Banach space $X$ has property (co), then for any scattered locally compact Hausdorff space $K$, the space $C_0(K,X)$ of continuous $X$-valued functions vanishing at infinity has…
σ-Continuous and Co-σ-continuous Maps
2009
In this chapter we isolate the topological setting that is suitable for our study. We first present 2.1–2.3 to follow an understandable logical scheme nevertheless the main contribution are presented in 2.4–2.7 and our main tool will be Theorem 2.32. An important concept will be the σ-continuity of a map Φ from a topological space (X, T) into a metric space (Y, g). The σ-continuity property is an extension of continuity suitable to deal with countable decompositions of the domain space X as well as with pointwise cluster points of sequences of functions Φn : X → Y, n = 1,2,… When (X,T) is a subset of a locally convex linear topological space we shall refine our study to deal with σ-slicely …
General Theory: Topological Aspects
2009
In Chapter 1, we have analyzed the structure of pip-spaces from the algebraic point of view only, (i.e., the compatibility relation). Here we will discuss primarily the topological structure given by the partial inner product itself. The aim is to tighten the definitions so as to eliminate as many pathologies as possible. The picture that emerges is reassuringly simple: Only two types of pip-spaces seem sufficiently regular to have any practical use, namely lattices of Hilbert spaces (LHS) or Banach spaces (LBS), that we have introduced briefly in the Introduction. Our standard reference on locally convex topological vector spaces (LCS) will be the textbook of Kothe [Kot69]. In addition, fo…
Dual attachment pairs in categorically-algebraic topology
2011
[EN] The paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category theory and universal algebra, and which is called categorically-algebraic (catalg) topology. The new framework is used to build a topological setting, based in a catalg extension of the set-theoretic membership relation "e" called dual attachment, thereby dualizing the notion of attachment introduced by the authors earlier. Following the recent interest of the fuzzy community in topological systems of S. Vickers, we clarify completely relationships between these structures and (dual) attachment, showing that unlike the former, the latter have no inh…
Categorical foundations of variety-based topology and topological systems
2012
The paper considers a new approach to fuzzy topology based on the concept of variety and developed in the framework of topological theories resembling those of Rodabaugh. As a result, a categorical generalization of the notion of topological system of Vickers is obtained, and its theory unfolded, which clarifies the relations between algebra and topology. We also justify the use of semi-quantales as the basic underlying structure for doing lattice-valued topology as well as provide a categorical framework incorporating the theory of bitopological spaces.
Structure of distributions generated by the scenery flow
2015
We expand the ergodic theory developed by Furstenberg and Hochman on dynamical systems that are obtained from magnifications of measures. We prove that any fractal distribution in the sense of Hochman is generated by a uniformly scaling measure, which provides a converse to a regularity theorem on the structure of distributions generated by the scenery flow. We further show that the collection of fractal distributions is closed under the weak topology and, moreover, is a Poulsen simplex, that is, extremal points are dense. We apply these to show that a Baire generic measure is as far as possible from being uniformly scaling: at almost all points, it has all fractal distributions as tangent …
Almost disjoint families of countable sets and separable complementation properties
2012
We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta $K_{\mathcal A}$ induced by almost disjoint families ${\mathcal A}$ of countable subsets of uncountable sets. For these spaces, we prove among others that $C(K_{\mathcal A})$ has the controlled variant of the separable complementation property if and only if $C(K_{\mathcal A})$ is Lindel\"of in the weak topology if and only if $K_{\mathcal A}$ is monolithic. We give an example of ${\mathcal A}$ for which $C(K_{\mathcal A})$ has the SCP, while $K_{\mathcal A}$ is not monolithic and an example of a space $C(K_{\mathcal A})$ with controlled and continuous SCP …