Francisco Javier Garcia-Pacheco

Universidad de Cádiz | UCA · Department of Mathematics

A geometric invariant or preserver is essentially a geometric property of the unit sphere of a real Banach space that remains invariant under the action of a surjective isometry onto the unit sphere of another real Banach space. A new geometric invariant of the unit ball of a real Banach space was introduced and analyzed in this manuscript: the cor...

The extremal structure of zero-neighbourhoods of a topological module is analyzed reaching unexpected conclusions when the module topology is not Hausdorff. These results motivate us to introduce the notion of metric modules, which are modules endowed with a translation-invariant metric, turning them into an (additive) topological group. We study t...

The intersection of all zero-neighborhoods in a topological module over a topological ring is a bounded and closed submodule whose inherited topology is the trivial topology. In this manuscript, we prove that this is the smallest closed submodule and thus replaces the null submodule in the Hausdorff setting. This fact motivates to introduce a new n...

An inner point of a non‐singleton convex set M$M$ is a point x∈M$x\in M$ satisfying that for all m∈M∖{x}$m\in M\setminus \lbrace x\rbrace$ there exists n∈M∖{m,x}$n\in M\setminus \lbrace m,x\rbrace$ such that x∈(m,n)$x\in (m,n)$. We prove the existence of convex compact subsets free of inner points in the infinite‐dimensional setting. Following our...

A real topological vector space is said to have the Krein–Milman property if every bounded, closed, convex subset has an extreme point. In the case of every bounded, closed, convex subset is the closed convex hull of its extreme points, then we say that the topological vector space satisfies the strong Krein–Milman property. The strong Krein–Milman...

Self-adjoint operators in smooth Banach spaces have been already defined in recent works. Here, we extend the concept of adjoint of an operator to the scope of (non-necessarily Hilbert) Banach spaces, obtaining in particular the notion of self-adjoint operator in the non-smooth case. As a consequence, we define the probability density operator on B...

Given a ring endowed with a ring order, we provide sufficient conditions for the order topology induced by the ring order to become a ring topology (analogous results for module orders are consequently derived). Finally, the notions of Radon and regular measures are transported to the scope of module-valued measures through module orders. Classical...

The index of strong rotundity is introduced. This index is used to determine how far an element of the unit sphere of a real Banach space is from being a strongly exposed point of the unit ball. This index is computed for Hilbert spaces. Characterizations of the set of rotund points and the set of smooth points are provided for a better understandi...

The stereographic projection is constructed in topological modules. Let A be an additively symmetric closed subset of a topological R-module M such that 0∈int(A). If there exists a continuous functional m*:M→R in the dual module M*, an invertible s∈U(R) and an element a in the topological boundary bd(A) of A in such a way that m*−1({s})∩int(A)=⌀, a...

The novel concept of focality is introduced for Borel probability measures on compact Hausdorff topological spaces. We characterize focal Borel probability measures as those Borel probability measures that are strictly positive on every nonempty open subset. We also prove the existence of focal Borel probability measures on compact metric spaces. L...

Chaotic and pathological phenomena in topological modules are studied in this manuscript. In particular, constructions of noncontinuous linear functionals are provided for a wide variety of topological modules. In addition, constructions of balanced and absorbing sets which are not neighborhoods of zero are also given in an extensive class of topol...

Effect algebras are the main object of study in quantum mechanics. Module measures are those measures defined on an effect algebra with values on a topological module. Let R be a topological ring and M a topological R-module. Let L be an effect algebra. The range of a module measure μ:L→M is studied. Among other results, we prove that if L is an sR...

For G∈Rm×n and g∈Rm, the minimization min∥Gψ−g∥2, with ψ∈Rn, is known as the Tykhonov regularization. We transport the Tykhonov regularization to an infinite-dimensional setting, that is min∥T(h)−k∥, where T:H→K is a continuous linear operator between Hilbert spaces H,K and h∈H,k∈K. In order to avoid an unbounded set of solutions for the Tykhonov r...

In this manuscript, we introduce the following novel concepts for real functions related to $ f $-convergence and $ f $-statistical convergence: $ f $-statistical continuity, $ f $-statistical derivative, and $ f $-strongly Cesàro derivative. In the first subsection of original results, the $ f $-statistical continuity is related to continuity. In...

The classical notion of statistical convergence has recently been transported to the scope of real normed spaces by means of the $ f $-statistical convergence for $ f $ a modulus function. Here, we go several steps further and extend the $ f $-statistical convergence to the scope of uniform spaces, obtaining particular cases of $ f $-statistical co...

The design of optimal Magnetic Resonance Imaging (MRI) coils is modeled as a minimum-norm problem (MNP), that is, as an optimization problem of the form $\min_{x\in\mathcal{R}}\|x\|$ min x ∈ R ∥ x ∥ , where $\mathcal{R}$ R is a closed and convex subset of a normed space X . This manuscript is aimed at revisiting MNPs from the perspective of Functio...

In this paper, we introduce the spaces of vector-valued sequences containing multiplier(weakly) statistically convergent series. The completeness of such spaces is studied as well as some relations between unconditionally convergent and weakly unconditionally Cauchy series of these spaces. We also obtain generalizations of some results regarding un...

Let $ T:X\to Y $ be a bounded linear operator between Banach spaces $ X, Y $. A vector $ x_0\in {\mathsf{S}}_X $ in the unit sphere $ {\mathsf{S}}_X $ of $ X $ is called a supporting vector of $ T $ provided that $ \|T(x_0)\| = \sup\{\|T(x)\|:\|x\| = 1\} = \|T\| $. Since matrices induce linear operators between finite-dimensional Hilbert spaces, we...

Every element in the boundary of the group of invertibles of a Banach algebra is a topological zero divisor. We extend this result to the scope of topological rings. In particular, we define a new class of semi-normed rings, called almost absolutely semi-normed rings, which strictly includes the class of absolutely semi-valued rings, and prove that...

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal poi...

Abstract Calculus: A Categorical Approach provides an abstract approach to calculus. It is intended for graduate students pursuing PhDs in pure mathematics but junior and senior researchers in basically any field of mathematics and theoretical physics will also be interested. Any calculus text for undergraduate students majoring in engineering, mat...

There are typically several perturbation methods for approaching the solution of weakly nonlinear vibrations (where the nonlinear terms are “small” compared to the linear ones): the Method of Strained Parameters, the Naive Singular Perturbation Method, the Method of Multiple Scales, the Method of Harmonic Balance and the Method of Averaging. The St...

A supporting vector of a matrix A for a certain norm \(\Vert \cdot \Vert \) on \(\mathbb {R}^n\) is a vector x such that \(\Vert x\Vert =1\) and \(\Vert Ax\Vert =\Vert A\Vert =\displaystyle \max _{\Vert y\Vert =1}\Vert Ay\Vert \). In this manuscript, we characterize the existence of supporting vectors in the infinite-dimensional case for both the \...

This manuscript determines the set of Pareto optimal solutions of certain multiobjective-optimization problems involving continuous linear operators defined on Banach spaces and Hilbert spaces. These multioptimization problems typically arise in engineering. In order to accomplish our goals, we first characterize, in an abstract setting, the set of...

The set of supporting vectors of a continuous linear operator \(T:X\rightarrow Y\) between normed spaces, denoted by \(\mathrm {suppv}(T)\) since 2017, is defined as \(\mathrm {suppv}(T):=\{x\in X:\Vert T(x)\Vert =\Vert T\Vert \Vert x\Vert \}\). In this manuscript, we study the lineability and coneability properties of \(\mathrm {suppv}(T)\), reach...

This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of $\mathbb{N} $ N and study the space of convergence associated with the filter. We notice that $c(X)$ c ( X ) is always a space of convergence associated with a filter (the Frechet filter); that if X is fin...

Inner structure appeared in the literature of topological vector spaces as a tool to characterize the extremal structure of convex sets. For instance, in recent years, inner structure has been used to provide a solution to The Faceless Problem and to characterize the finest locally convex vector topology on a real vector space. This manuscript goes...

The recently introduced concept of closed unit segment for rings allows to define convexity in modules. In this manuscript, we construct a new class of closed unit segments in totally ordered rings, which allows to define convex functions on the category of modules over totally ordered rings. Among other results, it is proved that the boundary of a...

The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the l...

Transcranial magnetic stimulation is a promising tool in neuroscience of which successful development is affected by the loud click noise originated when the stimulating coil is energized. This undesired sound is produced by the coil winding deformations generated by the Lorentz self-forces in the TMS device. Addressing the need for TMS systems tha...

Selfadjoint operators on Hilbert spaces have been extended to more general scopes such as certain real Banach spaces and complex Banach spaces endowed with a continuous Hermitian bilinear form. Here we propose a definition of selfadjoint operator that works for both real and complex Banach spaces and that naturally extends the classical concept of...

In this manuscript, we transport the classical Operator Theory on complex Banach spaces to normed modules over absolutely valued rings. In some cases, we are able to extend classical results on complex Banach spaces to normed modules over normed rings. In order to make sure that bounded linear maps on normed modules coincide with the continuous lin...

In this manuscript we provide an exact solution to the maxmin problem max ∥ A x ∥ subject to ∥ B x ∥ ≤ 1 , where A and B are real matrices. This problem comes from a remodeling of max ∥ A x ∥ subject to min ∥ B x ∥ , because the latter problem has no solution. Our mathematical method comes from the Abstract Operator Theory, whose strong machinery a...

In 1992, Kiendi, Adamy and Stelzner investigated under which conditions a certain type of function constituted a Lyapunov function for some time-invariant linear system. Six years later, it was obtained that this property holds if and only if the Banach space enjoys the self-extension property. However, the knowledge of these spaces needed to be ex...

The supporting vectors of a matrix A are the solutions of max∥x∥2=1∥Ax∥22. The generalized supporting vectors of matrices A1,⋯,Ak are the solutions of max∥x∥2=1∥A1x∥22+⋯+∥Akx∥22. Notice that the previous optimization problem is also a boundary element problem since the maximum is attained on the unit sphere. Many problems in Physics, Statistics and...

A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n ∈ N in X such that for every x ∈ X there exists a unique sequence of scalars ( λ n ) n ∈ N satisfying that x = ∑ n = 1 ∞ λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or co...

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Later on, they have been generalized to the context of real vector spaces by means of the inner points. Inner points can be seen as the most opposite concept to the extreme points. In this manuscript we solve the...

In this manuscript we provide an exact solution to the maxmin problem max Ax min Bx where A and B are real matrices. Our method comes from the Abstract Operator Theory, whose strong machinery allows us to reduce the previous problem to max Cx x ≤ 1 which can be solved exactly by relying on supporting vectors. Finally, as an appendix, we provide an...

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Very recently this concept was generalized to the one of inner points in the scope of vector spaces, which, among other things, allows to characterize the linear dimension of a vector space and also serves to pro...

We study measures defined on effect algebras. We characterize real-valued measures on effect algebras and find a class of effect algebras, that include the natural effect algebras of sets, on which σ -additive measures with values in a finite dimensional Banach space are always bounded. We also prove that in effect algebras the Nikodym and the Grot...

In this manuscript, we compute the Bishop-Phelps-Bollobás modulus for functionals in classical Banach spaces, such as Hilbert spaces, spaces of continuous functions c0 and ℓ1.

It has been recently proved that every real Banach space can be endowed with an equivalent norm in such a way that the new unit sphere contains a convex subset with non-empty interior relative to the unit sphere. In fact, under good conditions like separability or being weakly compactly generated, this renorming can be accomplished to have a dense...

Изучена геометрия единичного шара в пространстве $\ell_\infty(\Lambda)$ и в сопряженном пространстве и среди прочего доказано, что множество $\Lambda$ счетно тогда и только тогда, когда $1$ - выступающая точка шара $\mathsf{B}_{\ell_\infty(\Lambda)}$. С другой стороны, доказано, что множество $\Lambda$ конечно тогда и только тогда, когда функционал...

We study the geometry of the unit ball of ℓ∞(Λ) and of the dual space, proving, among other things, that Λ is countable if and only if 1 is an exposed point of \({B_{{\ell _\infty }\left( \Lambda \right)}}\). On the other hand, we prove that Λ is finite if and only if the δλ are the only functionals taking the value 1 at a canonical element and van...

We define the concepts of balanced set and absorbing set in modules over topological rings, which coincide with the usual concepts when restricting to topological vector spaces. We show that in a topological module over an absolute semi-valued ring whose invertibles approach 0, every neighborhood of 0 is absorbing. We also introduce the concept of...

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. In this manuscript we generalize the concept of internal point in real vector spaces by introducing a type of points, called inner points, that allows us to provide an intrinsic characterization of linear manifol...

A new method to obtain a continuous and fast measurement of light intensity is presented. It is targeted for Integrate and Fire pixels that pulse with a frequency proportional to illumination. The procedure is intended to speed up the pixel readout of low illuminated pixels. It does not require synchronisation of different digital signals, being co...

We prove that every infinite dimensional Banach space can be equivalently renormed so that the set of norm attaining functionals contains an infinite dimensional vector subspace.

This book pretends to compile the latest advances on vector-valued Banach limits as well as their applications to vector-valued almost convergence.

In this short note we prove that the Approximate Hyperplane Series property (AHSp) is hereditary to E-summands via characterizing the E-projections.

An inverse boundary element method and efficient optimisation techniques were combined to produce a versatile framework to design optimal TMS coils. The presented approach can be seen as an improvement and extension of the work introduced by Cobos Sanchez et al. [1] where the optimality of the resulting coil solutions was not guaranteed. This new n...

We prove that an absolute semi-valued ring is first-countable if the set of invertibles is separable and its closure contains 0. We also show that every linearly topologized topological module over an absolute semi-valued ring whose invertibles approach 0 has the trivial topology. We also show that every sequentially compact set in a topological mo...

The set of supporting vectors of a continuous linear operator, that is, the normalized vectors at which the operator attains its norm, is decomposed into its convex components. In the complex case, the set of supporting vectors of a nonzero functional is proved to be path-connected. We also introduce the concept of generalized supporting vectors fo...

A balanced and absorbing subset with empty interior has already been explicitly constructed in every normed space of dimension strictly greater than 1 (see [4]). However this construction varies depending whether the normed space is separable or not. In this note, we provide a unique construction by means of a family of balanced and absorbing sets...

Let (Formula presented.) be an isometric representation of a group (Formula presented.) in a Banach space (Formula presented.) over a normalizing non-discrete absolute valued division ring (Formula presented.). If (Formula presented.) and (Formula presented.) are supportive and (Formula presented.) verifies the separation property, then (Formula pr...

This paper is divided into four parts. In the first we study the existence of vector-valued Banach limits and show that a real Banach space with a monotone Schauder basis admits vector-valued Banach limits if and only if it is 1-complemented in its bidual. In the second we prove two vector-valued versions of Lorentz' intrinsic characterization of a...

We study some geometric properties related to the set obtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element ( h , k ) ∊ for H a Hilbert space.

Текст разделен на четыре части. В первой части мы изучаем существование векторнозначных банаховых пределов и показываем, что вещественное банахово пространство с монотонным базисом Шаудера допускает векторнозначные банаховы пределы тогда и только тогда, когда оно 1-дополняемо в своем втором сопряженном. Во второй части мы доказываем два векторнозна...

We prove that if a Banach space admits a biorthogonal system whose dual part is norming, then the set of norm-attaining functionals is lineable. As a consequence, if a Banach space admits a biorthogonal system whose dual part is bounded and its weak-star closed absolutely convex hull is a generator system, then the Banach space can be equivalently...

Our first result says that every real or complex infinite-dimensional normed space has an unbounded absolutely convex and absorbing subset with empty interior. As a consequence, a real normed space is finite-dimensional if and only if every convex subset containing 0 whose linear span is the whole space has non-empty interior. In our second result...

We introduce the concept of *-mapping as a selection of the duality mapping. We prove that the *-mappings are more general than the support mappings and provide a simple proof of the characterisation of smoothness by the norm to weak-star continuity of the *-mappings. As a consequence, we provide a characterisation of Hilbert spaces in terms of *-m...

In this manuscript we find another class of real Banach spaces which admit vector-valued Banach limits different from the classes found in [6, 7]. We also characterize the separating subsets of ℓ∞(X). For this we first need to study when the space of almost convergent sequences is closed in the space of bounded sequences, which turns out to happen...

We show that not all infinite dimensional closed subspaces of \({\ell_\infty}\) satisfy that the set of their norm-attaining functionals is lineable and prove that any closed subspace of \({\ell_\infty}\) containing c 0 has the property that the set of their norm-attaining functionals is lineable.

We characterize the points of $\left\|\cdot\right\|$-$w^*$ continuity of dual maps, turning out to be the smooth points. We prove that a Banach space has the Schur property if and only if it has the Dunford-Pettis property and there exists a dual map that is sequentially $w$-$w$ continuous at $0$. As consequence, we show the existence of smooth Ban...

It is shown that an infinite dimensional Banach space is hyper-barrelled if and only if it does not admit an infinite dimensional separable quotient.

In this manuscript we introduce a new class of convex sets called quasi-absolutely convex and show that a Hausdorff locally convex topological vector space satisfies the weak anti-proximinal property if and only if every totally anti-proximinal quasi-absolutely convex subset is not rare. This improves results from [7] and provides a partial positiv...

We gradually study i-operators on real vector spaces, on real topological vector spaces, and on real normed spaces. Among several things we prove the existence of real topological vector spaces (different from the James’ space) that are free of continuous i-operators. We also prove that every real normed space can be equivalently renormed to be fre...

Given a Banach space X, x is an element of S-X, and J(X)(x) = {x* is an element of S-X* : x*(x) = 1}, we define the set J(X)*(x) of all x* is an element of S-X* for which there exist two sequences (x(n))(n epsilon N) subset of S-X \ {x} and (x(n)*)(n is an element of N) subset of S-X* such that (x(n))(n is an element of N) converges to x, (x(n)*)(n...

New advances towards a (positive) solution to Ricceri’s (most famous) Conjecture are presented. One of these advances consists of showing that a totally anti-proximinal absolutely convex subset of a vector space is linearly open. We also prove that if every totally anti-proximinal convex subset of a vector space is linearly open then Ricceri’s Conj...

It is shown that, in a non-necessarily Hausdorff real topological vector space, if a subset is a countable disjoint union of convex sets closed in the subset, then those convex sets must be its convex components. On the other hand, by means of convex components we extend the notion of extreme point to non-convex sets, which entails a new equivalent...

The concepts of open unit ball and closed unit ball in a real or complex normed space are naturally extended to the scope of topological rings with unity. We then define a type of open (closed) sets called open (closed) unit neighborhoods of 0. We show among other things that in R and C the only non-trivial open and closed unit neighborhoods of 0 a...

The Banach-Mazur Conjecture for Rotations states that every transitive and separable Banach space must be a Hilbert space. The weak form of this conjecture states that every transitive and separable Banach space must be rotund. In this paper we show that every transitive Banach space in which all faces of the unit ball are invariant faces must be r...

Among other things it is shown that, in a dual isometric representation of a group on a rotund and smooth dual Banach space, the space of invariant vectors is 1-complemented.

In this manuscript we introduce the Universal Renorming, that is, a renorming technique which is universal in the sense that every equivalent norm on a real or complex normed space is determined by this technique. As an application of the universal renorming technique we prove that a large class of real Banach spaces can be equivalently renormed su...

In this paper we study the almost convergence and the almost summability in normed spaces. Among other things, spaces of sequences defined by the almost convergence and the almost summability are proved to be complete if the basis normed space is so. Finally, some classical properties such as completeness, reflexivity, Schur property, Grothendieck...

We give a new proof of the fact that equivalent norms on subspaces can be extended. This new proof is based on the Hahn-Banach Extension Theorem. We also give new characterizations for an equivalent norm on a dual space to be a dual norm. Finally, a new proof of a particular case of the Hahn-Banach Separation Theorem is provided without involving t...

In this brief communication we propose a vector-valued version of Lorentz’ intrinsic characterization of almost convergence, for which we find a legitimate extension of the concept of Banach limit to vector-valued sequences. Banach spaces 1-complemented in their biduals admit vector-valued Banach limits, whereas c 0 does not.

In this paper we continue a work that James started in 1971 about norm-attaining functionals on non-complete normed spaces by proving that every functional on a normed space is norm-attaining if and only if every proper,closed, convex subset with non-empty interior can be translated to have a non-zero, minimum-norm element. We also study this type...

It is proved that if X is infinite-dimensional, then there exists an infinitedimensional space of X-valued measures which have infinite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that ca(B; γX) n M, the measures with non-nite variation, contains a closed subspace. Other considerations concern the space...

In this manuscript we solve in the positive a question informally proposed by Enflo on the measure of the set of isometric reflection vectors in non-Hilbert 2-dimensional real Banach spaces. We also reformulate equivalently the separable quotient problem in terms of isometric reflection vectors. Finally, we give a new and easy example of a real Ban...

A totally anti-proximinal subset of a vector space is a non-empty proper subset which does not have a nearest point whatever is the norm that the vector space is endowed with. A Hausdorff locally convex topological vector space is said to have the (weak) anti-proximinal property if every totally anti-proximinal (absolutely) convex subset is not rar...

The Index of Rotundity Problem asks whether a Banach space which admits equivalent renormings with index of rotundity as small as desired also admits an equivalent rotund renorming. In this paper we continue the ongoing search for a negative answer to this question by making use of a new concept: asymptotically convex Banach spaces. Some applicatio...

Throughout this paper a study on the Krein–Milmam Property and the Bade Property is entailed reaching the following conclusions: If a real topological vector space satisfies the Krein–Milmam Property, then it is Hausdorff; if a real topological vector space satisfies the Krein–Milmam Property and is locally convex and metrizable, then all of its cl...

We construct, over many Banach spaces, infinite dimensional vector spaces of scalarly measurable functions that are not strongly measurable, and infinite dimensional vector spaces of ω * -scalarly measurable functions that are not scalarly measurable. A similar result will be proved for the set of McShane-integrable functions which are not Bochner-...

The purpose of this paper is to study certain geometrical properties for non-complete normed spaces. We show the existence of a non-rotund Banach space with a rotund dense maximal subspace. As a consequence, we prove that every separable Banach space can be renormed to be non-rotund and to contain a dense maximal rotund subspace. We then construct...

In this paper we consider the problem of the non-empty intersection of exposed faces in a Banach space. We find a sufficient condition to assure that the non-empty intersection of exposed faces is an exposed face. This condition involves the concept of inner point. Finally, we also prove that every minimal face of the unit ball must be an extreme p...

In this paper we study the geometry of isometric reflection vectors. In particular, we generalize known results by proving that the minimal face that contains an isometric reflection vector must be an exposed face. We also solve an open question by showing that there are isometric reflection vectors in any two dimensional subspace that are not isom...

Some classical Hahn–Schur Theorem-like results on the uniform convergence of unconditionally convergent series can be generalized to weakly unconditionally Cauchy series. In this paper, we obtain this type of generalization via a summability method based upon the concept of almost convergence. We also obtain a generalization of the main result in A...

We show that some pathological phenomena occur more often than one could expect, existing large algebraic structures (infinite dimensional vector spaces, algebras, positive cones or infinitely generated modules) enjoying certain special properties. In particular we construct infinite dimensional vector spaces of non-integrable, measurable functions...

In this short note we prove that every real Banach space with separable dual can be equivalently renormed so that the set of non-norm-attaining functionals on it is nowhere dense. This solves a question proposed by Enflo.

This article is divided into two parts. The first one is on the linear structure of the set of norm-attaining functionals on a Banach space. We prove that every Banach space that admits an infinite-dimensional separable quotient can be equivalently renormed so that the set of norm-attaining functionals contains an infinite-dimensional vector subspa...

The main result in this paper assures that, if X is a 2-dimensional real Banach space, then X is rotund if and only if every closed, connected subset of S X is of the form B X (x,r)∩S X .