Regular Open Subset Research Articles

A Point-Free Approach to Canonical Extensions of Boolean Algebras and Bounded Archimedean $$\ell$$-Algebras

Recently W. Holliday gave a choice-free construction of a canonical extension of a boolean algebra B as the boolean algebra of regular open subsets of the Alexandroff topology on the poset of proper filters of B. We make this construction point-free by replacing the Alexandroff space of proper filters of B with the free frame $$\mathcal {L}_B$$ generated by the bounded meet-semilattice of all filters of B (ordered by reverse inclusion) and prove that the booleanization of $$\mathcal {L}_B$$ is a canonical extension of B. Our main result generalizes this approach to the category $$\varvec{ ba \ell }$$ of bounded archimedean $$\ell$$ -algebras, thus yielding a point-free construction of canonical extensions in $$\varvec{ ba \ell }$$ . We conclude by showing that the algebra of normal functions on the Alexandroff space of proper archimedean $$\ell$$ -ideals of A is a canonical extension of $$A\in \varvec{ ba \ell }$$ , thus providing a generalization of the result of Holliday to $$\varvec{ ba \ell }$$ .

Order isomorphisms between bases of topologies

In this paper we will study the representations of isomorphisms between bases of topological spaces. It turns out that the perfect setting for this study is that of regular open subsets of complete metric spaces, but we have been able to show some results about arbitrary bases in complete metric spaces and also about regular open subsets in Hausdorff regular topological spaces.

Measure and Integration on Boolean Algebras of Regular Open Subsets in a Topological Space

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 content is a finitely additive measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such contents. We then explain the relationship between contents, residual charges, and Borel measures. We show that a content can be represented by a normal Borel measure, augmented with a liminal structure, which specifies how two or more regular open sets share the measure of their common boundary. In particular, a content on a locally compact Hausdorff space can be represented by a normal Borel measure and a liminal structure on the Stone-Čech compactification of that space. We also show how contents can be represented by Borel measures on the Stone space of the underlying Boolean algebra of regular open sets.

Subjective expected utility with imperfect perception

In many decisions under uncertainty, there are constraints on both the available information and the feasible actions. The agent can only make certain observations of the state space, and she cannot make them with perfect accuracy—she has imperfect perception. Likewise, she can only perform acts that transform states continuously into outcomes, and perhaps satisfy other regularity conditions. To incorporate such constraints, we modify the Savage decision model by endowing the state space S and outcome space X with topological structures. We axiomatically characterize a Subjective Expected Utility (SEU) representation of conditional preferences, involving a continuous utility function on X (unique up to positive affine transformations), and a unique probability measure on a Boolean algebra B of regular open subsets of S. We also obtain SEU representations involving a Borel measure on the Stone space of B — a “subjective” state space encoding the agent’s imperfect perception.

A Calculus of Regions Respecting Both Measure and Topology

Say that space is ‘gunky’ if every part of space has a proper part. Traditional theories of gunk, dating back to the work of Whitehead in the early part of last century, modeled space in the Boolean algebra of regular closed (or regular open) subsets of Euclidean space. More recently a complaint was brought against that tradition in Arntzenius (2008) and Russell (2008): Lebesgue measure is not even finitely additive over the algebra, and there is no countably additive measure on the algebra. Arntzenius advocated modeling gunk in measure algebras instead—in particular, in the algebra of Borel subsets of Euclidean space, modulo sets of Lebesgue measure zero. But while this algebra carries a natural, countably additive measure, it has some unattractive topological features. In this paper, we show how to construct a model of gunk that has both nice rudimentary measure-theoretic and topological properties. We then show that in modeling gunk in this way we can distinguish between finite dimensions, and that nothing in lost in terms of our ability to identify points as locations in space.

Subjective Expected Utility with Topological Constraints

In many decisions under uncertainty, there are technological constraints on the acts an agent can perform and on the events she can observe. To model this, we assume that the set S of possible states of the world and the set X of possible outcomes each have a topological structure. The only feasible acts are continuous functions from S to X, and the only observable events are regular open subsets of S. We axiomatically characterize Subjective Expected Utility (SEU) representations of conditional preferences over acts, involving a continuous utility function on X (unique up to positive affine transformations), and a unique Borel probability measure on S, along with an auxiliary apparatus called a liminal structure, which describes the agent’s imperfect perception of events. We also give other SEU representations, which use residual probability charges or compactifications of the state space.

Subjective Expected Utility with Imperfect Perception

In many decisions under uncertainty, there are constraints on both the available information and the feasible actions. The agent can only make certain observations of the state space, and she cannot make them with perfect accuracy — she has imperfect perception. Likewise, she can only perform acts that transform states continuously into outcomes, and perhaps satisfy other regularity conditions. To incorporate such constraints, we modify the Savage decision model by endowing the state space S and outcome space X with topological structures. We axiomatically characterize a Subjective Expected Utility (SEU) representation of conditional preferences, involving a continuous utility function on X (unique up to positive affine transformations), and a unique probability measure on a Boolean algebra B of regular open subsets of S. We also obtain SEU representations involving a Borel measure on the Stone space of B — a “subjective” state space encoding the agent’s imperfect perception.

DIMENSION DES FIBRES DE SPRINGER AFFINES POUR LES GROUPES

This article establishes a dimension formula for a group version of affine Springer fibers. We follow the method initiated by Bezrukavnikov in the case of Lie algebras. It consists in the introduction of a big enough regular open subset, with the same dimension as the affine Springer fiber. We show that, in the case of groups, such a regular open subset with analogous properties exists. Its construction needs the introduction of the Vinberg semi-group VG, for which we study an adjoint quotient χ+ and extend for χ+ the results previously established by Steinberg.

Global nonexistence for abstract evolution equations with dissipation

In this paper we study the initial value problem for the abstract evolution equations of the type [P(ut)]t+A(t,u(t))+Q(t,x,u,ut)=F(t,x,u),(t,x)∈J×Ω where 0<T≤∞, Ω is a bounded regular open subset of Rn. We establish the nonexistence result of global solutions with the initial energy controlled above by a critical value. We also give some applications concerning the classical equations. This improves earlier results in the literature.

Monotone complete C *-algebras and generic dynamics

Let S be the Stone space of a complete, non-atomic Boolean algebra. Let G be a countably infinite group of homeomorphisms of S. Let the action of G on S have a free dense orbit. Then we prove that, on a generic subset of S, the orbit equivalence relation coming from this action can also be obtained as an action of the (abelian) Dyadic Group. For the special case where the complete Boolean algebra is the algebra of regular open subsets of the real numbers, this reduces to a theorem of Sullivan-Weiss-Wright. By applying our new dynamical results we improve on an earlier paper by constructing a family of monotone complete C*-algebras, {B(t):t in T} with the following properties. First,T is large; it can be identified with the set of all subsets of the reals. Secondly each B(t) is a small C*-algebra,which is a monotone complete factor of Type III, is hyperfinite and is not a von Neumann algebra.Thirdly, when t is not equal to s, then B(s) is not isomorphic to B(t). In fact, B(s) and B(t) take different values in the classification semi-group for small monotone complete C*-algebras.

Regular opens in constructive topology and a representation theorem for overlap algebras

Giovanni Sambin has recently introduced the notion of an overlap algebra in order to give a constructive counterpart to a complete Boolean algebra. We propose a new notion of regular open subset within the framework of intuitionistic, predicative topology and we use it to give a representation theorem for (set-based) overlap algebras. In particular we show that there exists a duality between the category of set-based overlap algebras and a particular category of topologies in which all open subsets are regular.

The overlap algebra of regular opens

Overlap algebras are complete lattices enriched with an extra primitive relation, called “overlap”. The new notion of overlap relation satisfies a set of axioms intended to capture, in a positive way, the properties which hold for two elements with non-zero infimum. For each set, its powerset is an example of overlap algebra where two subsets overlap each other when their intersection is inhabited. Moreover, atomic overlap algebras are naturally isomorphic to the powerset of the set of their atoms. Overlap algebras can be seen as particular open (or overt) locales and, from a classical point of view, they essentially coincide with complete Boolean algebras. Contrary to the latter, overlap algebras offer a negation-free framework suitable, among other things, for the development of point-free topology. A lot of topology can be done “inside” the language of overlap algebra. In particular, we prove that the collection of all regular open subsets of a topological space is an example of overlap algebra which, under natural hypotheses, is atomless. Since they are a constructive counterpart to complete Boolean algebras and, at the same time, they have a more powerful axiomatization than Heyting algebras, overlap algebras are expected to turn out useful both in constructive mathematics and for applications in computer science.

A new $L^\infty$ estimate in optimal mass transport

Let $\Omega$ be a bounded Lipschitz regular open subset of $\mathbb {R}^d$ and let $\mu ,\nu$ be two probablity measures on $\overline {\Omega }$. It is well known that if $\mu =f dx$ is absolutely continuous, then there exists, for every $p>1$, a unique transport map $T_p$ pushing forward $\mu$ on $\nu$ and which realizes the Monge-Kantorovich distance $W_p(\mu ,\nu )$. In this paper, we establish an $L^\infty$ bound for the displacement map $T_p x-x$ which depends only on $p$, on the shape of $\Omega$ and on the essential infimum of the density $f$.

Sobolev spaces on Lie manifolds and regularity for polyhedral domains

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping properties of pseudodifferential operators. A tubular neighborhood theorem for Lie submanifolds allows us also to extend to regular open subsets of Lie manifolds the classical results on traces of functions in suitable Sobolev spaces. Our main application is a regularity result on polyhedral domains \mathbb P \subset \mathbb R^3 using the weighted Sobolev spaces \mathcal K^m_a(\mathbb P) . In particular, we show that there is no loss of \mathcal K^m_a –regularity for solutions of strongly elliptic systems with smooth coefficients. For the proof, we identify \mathcal K^m_a(\mathbb P) with the Sobolev spaces on \mathbb P associated to the metric r_{\mathbb P}^{-2} g_E , where g_E is the Euclidean metric and r_{\mathbb P}(x) is a smoothing of the Euclidean distance from x to the set of singular points of \mathbb P . A suitable compactification of the interior of \mathbb P then becomes a regular open subset of a Lie manifold. We also obtain the well-posedness of a non-standard boundary value problem on a smooth, bounded domain with boundary \mathcal O \subset \mathbb R^n using weighted Sobolev spaces, where the weight is the distance to the boundary.

Distributivity of the algebra of regular open subsets of [formula omitted

We compare the structure of the algebras P ( ω ) / fin and A ω / Fin , where A denotes the algebra of clopen subsets of the Cantor set. We show that the distributivity number of the algebra A ω / Fin is bounded by the distributivity number of the algebra P ( ω ) / fin and by the additivity of the meager ideal on the reals. As a corollary we obtain a result of A. Dow, who showed that in the iterated Mathias model the spaces β ω ∖ ω and β R ∖ R are not co-absolute. We also show that under the assumption t = h the spaces β ω ∖ ω and β R ∖ R are co-absolute, improving on a result of E. van Douwen.

Homogenization in general periodically perforated domains by a spectral approach

In this article we study the asymptotic behaviour as \(\epsilon\) tends to 0 of the Neumann problem $-\Delta u_\epsilon+u_\epsilon=\( in a \)\epsilon$-periodic bounded open set \(\Omega_\epsilon\) of \({\mathbb R}^d\). The period cell of \(\Omega_\epsilon\) is equal to \(\epsilon Y_\epsilon\) where \(Y_\epsilon\) is a regular open subset of the d-dimensional torus. We prove that if there exists a smallest integer \(n\geq 1\) such that the n-th non-zero eigenvalue \(\Lambda_n(\epsilon)\) of the spectral problem \(-\Delta V_\epsilon=\Lambda(\epsilon) V_\epsilon\) in \(Y_\epsilon\) satisfies \(\Lambda_n(\epsilon) >> \epsilon^2\), the limiting problem is a linear system of second order p.d.e.'s, of size n. By this spectral approach we extend in the periodic framework a result due to Khruslov without making strong geometrical assumptions on the perforated domain \(\Omega_\epsilon\).

A density result in Sobolev spaces

We prove, for 1⩽ p<∞ and Ω a polygonal or regular open subset of R N , the density in W 1,p(Ω) of a set of regular functions satisfying a homogeneous Neumann condition on the boundary of Ω. We also give applications of this result and a generalization to mixed Dirichlet–Neumann boundary conditions.

Embedding compact spaces into compact spaces with few regular open Baire subsets

It is shown that every compact space X can be embedded as a retract of a compact space Y so that every regular open Baire subset of Y is clopen. Furthermore, if X is a continuum, Y is a continuum as well and Y has only two regular open Baire subsets. This extends a result of V.V. Fedorčuk.

Order topology in vector lattices with closure by one step

An order topology in vector lattices and Boolean algebras is studied under the additional condition of “closure by one step” that generalizes the well-known “regularity” property of Boolean algebras and K-spaces. It is proved that in a vector lattice or a Boolean algebra possessing such a property there exists a basis of solid neighborhoods of zero with respect to an order topology. An example of a Boolean algebra without basis of solid neighborhoods of zero (an algebra of regular open subsets of the interval (0, 1)) is given. Bibliography: 3 titles.

Equational compactness of bi-frames and projection algebras

We generalize D. Kelly's and K. A. Nauryzbaev's results of 1-variable and 2-variable equational compactness of complete distributive lattices satisfying the infinite distributive law and its dual (“bi-frames”) to objects similar to monadic algebras (which we will callprojection algebras). This will lead us in particular to an example of bi-frame that is not 3-variable equationally compact, even forcountable equation systems, thus solving a problem of G. Gratzer. This example is realized as a certain complete sublattice of the complete Boolean algebra of regular open subsets of some Polish space.