scholarly journals Extensions of ordered theories by generic predicates

2013 ◽  
Vol 78 (2) ◽  
pp. 369-387 ◽  
Author(s):  
Alfred Dolich ◽  
Chris Miller ◽  
Charles Steinhorn
Keyword(s):  
Open Subset ◽  
Connected Components ◽  
Linear Orders ◽  
Algebraic Numbers ◽  
Borel Set ◽  
Cartesian Power ◽  
Open Set ◽  
Unary Relation

Given a theory T extending that of dense linear orders without endpoints (DLO), in a language ℒ ⊇ {<}, we are interested in extensions T′ of T in languages extending ℒ by unary relation symbols that are each interpreted in models of T′ as sets that are both dense and codense in the underlying sets of the models.There is a canonically “wild” example, namely T = Th(〈ℝ, <, +, ·〉) and T′ = Th(〈ℝ, <, +, · ℚ 〉). Recall that T is o-minimal, and so every open set definable in any model of T has only finitely many definably connected components. But it is well known that 〈ℝ, <, +, · ℚ 〉 defines every real Borel set, in particular, every open subset of any finite cartesian power of ℝ and every subset of any finite cartesian power of ℚ. To put this another way, the definable open sets in models of T are essentially as simple as possible, while T′ has a model where the definable open sets are as complicated as possible, as is the structure induced on the new predicate.In contrast to the preceding example, if ℝalg is the set of real algebraic numbers and T′ Th(〈ℝ, <, +, ·, 〈alg〉), then no model of T′ defines any open set (of any arity) that is not definable in the underlying model of T.

On topological properties of weakly m-convex sets

2021 ◽  
Vol 34 ◽  
pp. 75-84
Author(s):  
Tetiana Osipchuk
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Topological Classification ◽  
Connected Components ◽  
Topological Properties ◽  
Convex Domains ◽  
Closed Set ◽  
Dimensional Plane ◽  
Open Set ◽  
Plane Passing

The topological properties of classes of generally convex sets in multidimensional real Euclidean space $\mathbb{R}^n$, $n\ge 2$, known as $m$-convex and weakly $m$-convex, $1\le m<n$, are studied in the present work. A set of the space $\mathbb{R}^n$ is called \textbf{\emph{$m$-convex}} if for any point of the complement of the set to the whole space there is an $m$-dimensional plane passing through this point and not intersecting the set. An open set of the space is called \textbf{\emph{weakly $m$-convex}}, if for any point of the boundary of the set there exists an $m$-dimensional plane passing through this point and not intersecting the given set. A closed set of the space is called \textbf{\emph{weakly $m$-convex}} if it is approximated from the outside by a family of open weakly $m$-convex sets. These notions were proposed by Professor Yuri Zelinskii. It is known the topological classification of (weakly) $(n-1)$-convex sets in the space $\mathbb{R}^n$ with smooth boundary. Each such a set is convex, or consists of no more than two unbounded connected components, or is given by the Cartesian product $E^1\times \mathbb{R}^{n-1}$, where $E^1$ is a subset of $\mathbb{R}$. Any open $m$-convex set is obviously weakly $m$-convex. The opposite statement is wrong in general. It is established that there exist open sets in $\mathbb{R}^n$ that are weakly $(n-1)$-convex but not $(n-1)$-convex, and that such sets consist of not less than three connected components. The main results of the work are two theorems. The first of them establishes the fact that for compact weakly $(n-1)$-convex and not $(n-1)$-convex sets in the space $\mathbb{R}^n$, the same lower bound for the number of their connected components is true as in the case of open sets. In particular, the examples of open and closed weakly $(n-1)$-convex and not $(n-1)$-convex sets with three and more connected components are constructed for this purpose. And it is also proved that any compact weakly $m$-convex and not $m$-convex set of the space $\mathbb{R}^n$, $n\ge 2$, $1\le m<n$, can be approximated from the outside by a family of open weakly $m$-convex and not $m$-convex sets with the same number of connected components as the closed set has. The second theorem establishes the existence of weakly $m$-convex and not $m$-convex domains, $1\le m<n-1$, $n\ge 3$, in the spaces $\mathbb{R}^n$. First, examples of weakly $1$-convex and not $1$-convex domains $E^p\subset\mathbb{R}^p$ for any $p\ge3$, are constructed. Then, it is proved that the domain $E^p\times\mathbb{R}^{m-1}\subset\mathbb{R}^n$, $n\ge 3$, $1\le m<n-1$, is weakly $m$-convex and not $m$-convex.

scholarly journals Almost-continuous path connected spaces

1981 ◽  
Vol 4 (4) ◽  
pp. 823-825
Author(s):  
Larry L. Herrington ◽  
Paul E. Long
Keyword(s):  
Continuous Function ◽  
Connected Components ◽  
Continuous Path ◽  
Open Set ◽  
Regular Open ◽  
Path Connected ◽  
Almost Continuous ◽  
Connected Spaces

M. K. Singal and Asha Rani Singal have defined an almost-continuous functionf:X→Yto be one in which for eachx∈Xand each regular-open setVcontainingf(x), there exists an openUcontainingxsuch thatf(U)⊂V. A spaceYmay now be defined to be almost-continuous path connected if for eachy0,y1∈Ythere exists an almost-continuousf:I→Ysuch thatf(0)=y0andf(1)=y1An investigation of these spaces is made culminating in a theorem showing when the almost-continuous path connected components coincide with the usual components ofY.

scholarly journals THE CLOSED RANGE PROPERTY FOR BANACH SPACE OPERATORS

2008 ◽  
Vol 50 (1) ◽  
pp. 17-26 ◽  
Author(s):  
THOMAS L. MILLER ◽  
VLADIMIR MÜLLER
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Open Subset ◽  
Weak Topology ◽  
Bounded Operator ◽  
Complex Banach Space ◽  
Fréchet Space ◽  
Closed Range ◽  
Open Set ◽  
Banach Space Operators

AbstractLetTbe a bounded operator on a complex Banach spaceX. LetVbe an open subset of the complex plane. We give a condition sufficient for the mappingf(z)↦ (T−z)f(z) to have closed range in the Fréchet spaceH(V,X) of analyticX-valued functions onV. Moreover, we show that there is a largest open setUfor which the mapf(z)↦ (T−z)f(z) has closed range inH(V,X) for allV⊆U. Finally, we establish analogous results in the setting of the weak–* topology onH(V, X*).

Descriptive sets

1966 ◽  
Vol 291 (1426) ◽  
pp. 353-367 ◽  
Keyword(s):  
Metric Space ◽  
Regular Space ◽  
Countable Union ◽  
Borel Set ◽  
Borel Sets ◽  
Countable Intersection ◽  
Completely Regular ◽  
Closed Sets ◽  
Open Set

A theory of descriptive Baire sets is developed for an arbitrary completely regular space. It is shown that descriptive Baire sets are Baire sets and that they form a system closed under countable union, countable intersection and intersection with a Baire set. If a descriptive Borel set (Rogers 1965) is a Baire set then it is a descriptive Baire set. If every open set is a countable union of closed sets, the descriptive Baire sets coincide with the descriptive Borel sets. It follows, in particular, that in a metric space a set is descriptive Baire, if, and only if, it is absolutely Borel and Lindelöf.

Locally Convex Hypersurfaces

1973 ◽  
Vol 25 (3) ◽  
pp. 531-538 ◽  
Author(s):  
L. B. Jonker ◽  
R. D. Norman
Keyword(s):  
Convex Body ◽  
Open Subset ◽  
Convex Hypersurface ◽  
Topological Manifold ◽  
Continuous Map ◽  
Open Set ◽  
Locally Convex ◽  
Convex Hypersurfaces

Let M be an n-dimensional connected topological manifold. Let ξ : M → Rn+1 be a continuous map with the following property: to each x ∈ M there is an open set x ∈ Ux ⊂ M, and a convex body Kx ⊂ Rn+1 such that ξ(UX) is an open subset of ∂Kx and such that is a homeomorphism onto its image. We shall call such a mapping ξ a locally convex immersion and, along with Van Heijenoort [8] we shall call ξ(M) a locally convex hypersurface of Rn+1.

scholarly journals Behnke-Stein theorem on complex spaces with singularities

1995 ◽  
Vol 137 ◽  
pp. 183-194 ◽  
Author(s):  
M. Colţoiu ◽  
A. Silva
Keyword(s):  
Riemann Surface ◽  
Open Subset ◽  
Classical Result ◽  
Compact Riemann Surface ◽  
Connected Components ◽  
Restriction Map ◽  
Complex Spaces ◽  
Holomorphic Approximation ◽  
Dense Image

Let X be a non-compact Riemann surface and D ⊂ X an open subset. By a classical result due to Behnke and Stein [2] D is Runge in X (i.e. the restriction map has dense image) iff X\D has no compact connected components. In other words the obstruction to holomorphic approximation is purely topological. This result has been generalized to 1-dimensional Stein spaces by Mihalache in [11].

Convolution in perfect Lie groups

2016 ◽  
Vol 161 (1) ◽  
pp. 31-45
Author(s):  
YVES BENOIST ◽  
NICOLAS DE SAXCÉ
Keyword(s):  
Hausdorff Dimension ◽  
Lie Groups ◽  
Haar Measure ◽  
Lie Group ◽  
Borel Measure ◽  
Absolutely Continuous ◽  
Borel Set ◽  
Open Set ◽  
Convolution Power ◽  
Smooth Density

AbstractLetGbe a connected perfect real Lie group. We show that there exists α < dimGandp∈$\mathbb{N}$* such that if μ is a compactly supported α-Frostman Borel measure onG, then thepth convolution power μ*pis absolutely continuous with respect to the Haar measure onG, with arbitrarily smooth density. As an application, we obtain that ifA⊂Gis a Borel set with Hausdorff dimension at least α, then thep-fold product setApcontains a non-empty open set.

scholarly journals A note on a family of surfaces with $$p_g=q=2$$ and $$K^2=7$$

2021 ◽  
Author(s):  
Matteo Penegini ◽  
Roberto Pignatelli
Keyword(s):  
Open Subset ◽  
Moduli Space ◽  
General Type ◽  
Connected Components ◽  
Surfaces Of General Type ◽  
Alternative Construction ◽  
Albanese Map

AbstractWe study a family of surfaces of general type with $$p_g=q=2$$ p g = q = 2 and $$K^2=7$$ K 2 = 7 , originally constructed by C. Rito in [35]. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus $$\mathcal {M}$$ M in the moduli space of surfaces of general type. In particular we prove that $$\mathcal {M}$$ M is an open subset, and it has three connected components, all of which are 2-dimensional, irreducible and generically smooth

Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces

2011 ◽  
Vol 63 (2) ◽  
pp. 436-459 ◽  
Author(s):  
Kotaro Mine ◽  
Katsuro Sakai
Keyword(s):  
Simplicial Complex ◽  
Open Subset ◽  
Simplicial Complexes ◽  
Locally Finite ◽  
Direct Sum ◽  
Open Set ◽  
Finite Dimensional ◽  
Metric Topology ◽  
Dimensional Simplicial Complex

Abstract Let F be a non-separable LF-space homeomorphic to the direct sum , where . It is proved that every open subset U of F is homeomorphic to the product |K| × F for some locally finite-dimensional simplicial complex K such that every vertex v ∈ K(0) has the star St(v, K) with card St(v, K)(0) < 𝒯 = sup 𝒯n (and card K(0) ≤ 𝒯 ), and, conversely, if K is such a simplicial complex, then the product |K| × F can be embedded in F as an open set, where |K| is the polyhedron of K with the metric topology.

THERMIC MINORANTS AND REDUCTIONS OF SUPERTEMPERATURES

2015 ◽  
Vol 99 (1) ◽  
pp. 128-144 ◽  
Author(s):  
NEIL A. WATSON
Keyword(s):  
Open Subset ◽  
Lower Semicontinuous ◽  
Open Set

Let $u$ be a supertemperature on an open set $E$, and let $v$ be a related temperature on an open subset $D$ of $E$. For example, $v$ could be the greatest thermic minorant of $u$ on $D$, if it exists. Putting $w=u$ on $E\setminus D$ and $w=v$ on $D$, we investigate whether $w$, or its lower semicontinuous smoothing, is a supertemperature on $E$. We also give a representation of the greatest thermic minorant on $E$, if it exists, in terms of PWB solutions on an expanding sequence of open subsets of $E$ with union $E$.  In addition, in the case of a nonnegative supertemperature, we prove inequalities that relate reductions to Dirichlet solutions. We also prove that the value of any reduction at a given time depends only on earlier times.

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