This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The motivation arises from studying random elements talking values in N.B.A.
Those topological properties are important assumptions that enable us to avoid possible difficulties when generalising concepts of random variable convergence, the definition of conditional law and others.
For each N.B.A., there exists a finite measure space $(E, {\mathcal E}, μ)$ such that the N.B.A. is isomorphic to $(\widetilde{\mathcal E}, \widetildeμ)$ resulting from the factorisation of initial $σ$-algebra by the ideal of negligible sets. We focus on topological properties of $(\widetilde{\mathcal E}, \widetildeμ)$ in general setting when $μ$ can be an infinite measure. In case when $μ$ is infinite, we also consider properties of $\widetilde{\mathcal E}_{fin} \subseteq \widetilde{\mathcal E}$ consisting of classes of measurable sets having finite measure.
The compactness and separability of the N.B.A. are characterised using the newly defined terms of approximability and uniform approximability of the corresponding measure space.
Finally, conditions on $(E,\mathcal E,μ)$ are derived for separability and compactness of $\widetilde{\mathcal E}$ and $\widetilde{\mathcal E}_{fin}.$
A duality of $κ$-normed topological vector spaces is defined and investigated. For such spaces the analog of the Mackey-Arens theorem is proved. There are investigated cases, when $κ$-normability of a topological vector space implies its local convexity. There are given applications of $κ$-normed spaces for resolutions of differential equations and for approximations of functions in mathematical economy.
Relying on Kolmogorov's classical characterization of normable Topological Vector spaces, we study the normability of those Probabilistic Normed Spaces that are also Topological Vector spaces and provide a characterization of normable Šerstnev spaces. We also study the normability of other two classes of Probabilistic Normed Spaces.
In this paper, we define precompact set in intuitionistic fuzzy metric spaces and prove that any subset of an intuitionistic fuzzy metric space is compact if and only if it is precompact and complete. Also we define topologically complete intuitionistic fuzzy metrizable spaces and prove that any $G_{δ}$ set in a complete intuitionistic fuzzy metric spaces is a topologically complete intuitionistic fuzzy metrizable space and vice versa. Finally, we define intuitionistic fuzzy normed spaces and fuzzy boundedness for linear operators and so we prove that every finite dimensional intuitionistic fu…
In this paper, we approach the question if some of the separation axioms are equivalent in the class of asymmetric normed spaces. In particular, we make a remark on a known theorem which states that every $T_1$ asymmetric normed space with compact closed unit ball must be finite-dimensional. We also explore the product structure of these spaces and characterize the topological (covering) dimension of all finite-dimensional asymmetric normed spaces.
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 $(X,weak)$…
Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis is placed on the application of set-theoretic methods to the study of Boolean topological groups.
The regular open subsets of a topological space form a Boolean algebra, where the `join' of two regular open sets is the interior of the closure of their union. A `credence' is a finitely additive probability measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such credences. We then explain the relationship between credences, residual charges, and Borel probability measures. We show that a credence can be represented by a normal Borel measure, augmented with a `liminal structure', which specifies how two or more regular open sets share the pro…
Topological groups whose underlying spaces are basically disconnected, $F$-, or $F'$-spaces but not $P$-spaces are considered. It is proved, in particular, that the existence of a Lindelöf basically disconnected topological group which is not a $P$-space is equivalent to the existence of a Boolean basically disconnected Lindelöf group of countable pseudocharacter, that free and free Abelian topological groups of zero-dimensional non-$P$-spaces are never $F'$-spaces, and that the existence of a free Boolean $F'$-group which is not a $P$-space is equivalent to that of selective ultrafilters on $…
Given a partially ordered set $P$ there exists the most general Boolean algebra $F(P)$ which contains $P$ as a generating set, called the {\it free Boolean algebra} over $P$. We study free Boolean algebras over posets of the form $P=P_0\cup P_1$, where $P_0,P_1$ are well orderings. We call them {\it nearly ordinal algebras}. Answering a question of Maurice Pouzet, we show that for every uncountable cardinal $κ$ there are $2^κ$ pairwise non-isomorphic nearly ordinal algebras of cardinality $κ$. Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lat…
We compare the forcing related properties of a complete Boolean algebra B with the properties of the convergences $λ_s$ (the algebraic convergence) and $λ_{ls}$ on B generalizing the convergence on the Cantor and Aleksandrov cube respectively. In particular we show that $λ_{ls}$ is a topological convergence iff forcing by B does not produce new reals and that $λ_{ls}$ is weakly topological if B satisfies condition $(\hbar)$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $λ_{ls}$ is a weakly topological convergence, then B is a $2^{\mathfrak h}$-cc algebra or in some generic extensi…
We answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density pi(A), resp. the topological density d(A), of a Boolean algebra A commutes with formation of ultraproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra.
A topological commutative ring is said to be rigid when for every set $X$, the topological dual of the $X$-fold topological product of the ring is isomorphic to the free module over $X$. Examples are fields with a ring topology, discrete rings, and normed algebras. Rigidity translates into a dual equivalence between categories of free modules and of "topologically-free" modules and, with a suitable topological tensor product for the latter, one proves that it lifts to an equivalence between monoids in this category (some suitably generalized topological algebras) and coalgebras. In particular,…
A Boolean algebra $\mathcal A$ has the Nikodym property if every pointwise bounded sequence of bounded finitely additive measures on $\mathcal A$ is uniformly bounded. Assuming the Diamond Principle $\Diamond$, we will construct an example of a minimally generated Boolean algebra $\mathcal A$ with the Nikodym property. The Stone space of such an algebra must necessarily be an Efimov space. The converse is, however, not true - again under $\Diamond$ we will provide an example of a minimally generated Boolean algebra whose Stone space is Efimov but which does not have the Nikodym property. The r…
We study a two-parameter generalization of the Freese-Nation Property of boolean algebras and its order-theoretic and topological consequences. For every regular infinite κ, the (κ,κ)-FN, the (κ^+,κ)-FN, and the κ-FN are known to be equivalent; we show that the family of properties (λ,μ)-FN for λ>μform a true two-dimensional hierarchy that is robust with respect to coproducts, retracts, and the exponential operation. The (κ,\aleph_0)-FN in particular has strong consequences for base properties of compacta (stronger still for homogeneous compacta), and these consequences have natural duals in t…
Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion of any abstract algebra is the extended Stone dual space of the Boolean algebra of recognizable subsets of the abstract algebra endowed with certain residuation operations. These results identify a connection between topological algebra as applied in algebra and Stone duality as applied in logic, and show that the notion of recognition originating in compute…
A fundamental result proved by Bourgain, Fremlin and Talagrand states that the space $B_1(M)$ of Baire one functions over a Polish space $M$ is an angelic space. Stegall extended this result by showing that the class $B_1(M,E)$ of Baire one functions valued in a normed space $E$ is angelic. These results motivate our study of various topological properties in the classes $B_α(X,G)$ of Baire-$α$ functions, where $α$ is a nonzero countable ordinal, $G$ is a metrizable non-precompact abelian group and $X$ is a $G$-Tychonoff first countable space. In particular, we show that (1) $B_α(X,G)$ is a $κ…
We give a short proof of an improved version of the Effros Open Mapping Principle via a shift-compactness theorem (also with a short proof), involving `sequential analysis' rather than separability, deducing it from the Baire property in a general Baire-space setting (rather than under topological completeness). It is applicable to absolutely-analytic normed groups (which include complete metrizable topological groups), and via a Steinhaus-type Sum-set Theorem (also a consequence of the shift-compactness theorem) includes the classical Open Mapping Theorem (separable or otherwise).
We show that for a normal locally-${\mathscr P}$ space $X$ (where ${\mathscr P}$ is a topological property subject to some mild requirements) the subset $C_{\mathscr P}(X)$ of $C_b(X)$ consisting of those elements whose support has a neighborhood with ${\mathscr P}$, is a subalgebra of $C_b(X)$ isometrically isomorphic to $C_c(Y)$ for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone--Čech compactification $βX$ of $X$ and contains $X$ as a dense subspace. Under certain conditions, $C_{\mathscr P}(X)$ coinci…
The notions of a {\em 2-precontact space}\/ and a {\em 2-contact space}\/ are introduced. Using them, new representation theorems for precontact and contact algebras are proved. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. As applications of the obtained results, we get new connected versions of the Stone Duality Theorems for Boolean algebras and for complete Boolean algebras, as well as a Smirnov-type theorem for a kind of compact $T_0$-extensions of compact Hausdorff extremally disconnected spaces. We also introduce the notion …
It is proved that the existence of a countable extremally disconnected Boolean topological group containing a family of open subgroups whose intersection has empty interior implies the existence of a rapid ultrafilter.
We prove that colouring of pairs from aleph_2 with strong properties exists. The easiest to state (and quite a well known problem) it solves: there are two topological spaces with cellularity aleph_1 whose product has cellularity aleph_2 ; equivalently we can speak on cellularity of Boolean algebras or on Boolean algebras satisfying the aleph_2-c.c. whose product fails the aleph_2-c.c. We also deal more with guessing of clubs.
A topological space $X$ is defined to have an $ω^ω$-base if at each point $x\in X$ the space $X$ has a neighborhood base $(U_α[x])_{α\inω^ω}$ such that $U_β[x]\subset U_α[x]$ for all $α\leβ$ in $ω^ω$. We characterize topological and uniform spaces whose free (locally convex) topological vector spaces or free (Abelian or Boolean) topological groups have $ω^ω$-bases.
A denumerable cellular family of a topological space $\mathbf{X}$ is an infinitely countable collection of pairwise disjoint non-empty open sets of $ \mathbf{X}$. It is proved that the following statements are equivalent in $\mathbf{ZF}$: (i) For every infinite set $X,[X]^{<ω}\mathbf{\ }$has a denumerable subset. (ii) Every infinite $0$-dimensional Hausdorff space admits a denumerable cellular family. It is also proved that (i) implies the following: (iii) Every infinite Hausdorff Baire space has a denumerable cellular family. Among other results, the following theorems are also proved in $\ma…
For given Boolean algebras $\mathbb{A}$ and $\mathbb{B}$ we endow the space $\mathcal{H}(\mathbb{A},\mathbb{B})$ of all Boolean homomorphisms from $\mathbb{A}$ to $\mathbb{B}$ with various topologies and study convergence properties of sequences in $\mathcal{H}(\mathbb{A},\mathbb{B})$. We are in particular interested in the situation when $\mathbb{B}$ is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on $\mathbb{A}$ in random extensions of the set-theoretical universe. This appears to have strong connectio…
