Almeida, R

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Repositório Institucional da Universidade de Aveiro

Connectedness and Compactness on Standard SetsRicardo Almeidaricardo.almeida@ua.ptDepartment of Mathematics, University of Aveiro, 3810-193 Aveiro, PortugalAbstractWe present a nonstandard characterization of connected compact setsMathematics Subject Classification 2010: 26E35, 54D05, 54D30.Keywords: Nonstandard analysis, Connected set, Compact sets1 IntroductionThere is as yet no simple nonstandard characterization of connectedness, and little work has beendone in that direction. In [4], Steven Leth presents a sufficient condition for a set A ⊆ Rn to beconnected. As Leth remarked, however, it is not a necessary condition. His hypothesis involvesinternal polygonal paths joining distinct points. We will work with hyper-finite sets instead ofpolygonal paths, thus eliminating the implicit local path-connectedness that is present in [4]. Wemention also the work of Se´rgio Rodrigues [5] characterizing connectedness in nonstandard termsusing the monad of a set.The paper is organized as follows. In section 2 we collect some necessary background for thereader’s convenience. In section 3 we present new results about connectedness and compactnesson standard sets; we introduce a new concept, the discrete infinitesimal path, which will be usedto characterize connected compact sets in metric spaces.2 PreliminariesAll the sets which come up in classical analysis have nonstandard extensions using a map denotedby ”∗”. For example, if R denotes the set of real numbers, ∗R will be its nonstandard extension.This extension contains ”ideal elements”, like infinitesimals and infinite numbers, but also a copyof the set of real numbers, denoted by the symbol σR. It is not our intention to give a fullexposition on this subject, we will just fix notation and present some results needed. For furtherdetails, the reader is referred to [2, 3, 6, 8].Definition 1. Let x, y ∈ ∗R. We say that1. x is infinitesimal if |x| < , for all positive real number  and we write x ≈ 0;2. x is finite if, for some positive real number , |x| < ;3. x is infinite (or infinitely large) if it is not finite, i.e., for any positive real number , |x| > ;we write x ≈ ∞;4. x, y are infinitely close if x− y is infinitesimal; we write x ≈ y.In the following, (X, d) is a metric space.1Definition 2. For x ∈ ∗X, the monad of x is the subset of ∗X given byµ(x) := {y ∈ ∗X | d(x, y) ≈ 0}.As before, the nonstandard extension of X contains a copy of the original set, which we denoteby σX (elements of σX are called standard). A point y ∈ ∗X is nearstandard if there exists somestandard x ∈ σX such that y ∈ µ(x); in this case we say that x is the standard part of y and writest(y) = x. We say that x, y ∈ ∗X are infinitely close, and write x ≈ y, if d(x, y) ≈ 0. If x and yare not infinitely close, we write x 6≈ y.The set of the nearstandard points of ∗X isns(∗X) :=⋃{µ(x) |x ∈ σX}.Theorem 1. [3] Let A ⊆ X. Then1. A is open if and only if for all a ∈ σA, µ(a) ⊆ ∗A holds;2. A is closed if and only if, whenever a ∈ ∗A and a ≈ x for some x ∈ σX, we have x ∈ σA;3. A is compact if and only if for all a ∈ ∗A, there is an x ∈ σA with a ≈ x;In every metric space, monads of distinct standard points are disjoint (see [3]). Therefore, forall x ∈ ns(∗X), there exists exactly one element in σX, called st(x), infinitely close to x. Hencewe have a well-defined functionst : ns(∗X) → σXx 7→ st(x)called the standard part function.Theorem 2. [3] Let X and Y be two topological spaces and f : X → Y a function. Then f iscontinuous if and only if∀x ∈ σX f(µ(x)) ⊆ µ(f(x)),or equivalently,∀x ∈ σX ∀y ∈ ∗X [x ≈ y ⇒ f(x) ≈ f(y)].3 Main resultsIn what follows, (X, dX) and (Y, dY ) will denote two metric spaces, and A ⊆ X a nonemptysubset. To simplify notation, we will denote both metrics by the same symbol d. Given two pointsx, y ∈ ∗A, we define the set (possibly external)P∗Ax,y := {u = (un)n=1,...,N |N ∈ ∗N, u1 = x, uN = y, un ∈ ∗Aand un ≈ un+1, for all n ∈ {1, . . . , N − 1}}.We call the hyper-finite sequence u = (un)n∈{1,...N} a discrete infinitesimal path (abbreviationd.i.p.) joining x to y in ∗A. We define a binary relation on ∗A by x ∼ y if P∗Ax,y is nonempty; it iseasy to prove that ∼ is an equivalence relation.We will simply write Px,y instead of P∗Ax,y whenever there is no danger of confusion.The existence of (standard) discrete paths joining points on connected sets is known. In fact,it can be proved that if A is a connected set and  is a (standard) real, then for all x, y ∈ A, thereexists a finite sequence of points, all lying in A,x = u1, u2, . . . , un = y,such that the distance between any two successive points in this sequence is less than .The next result follows from the compactness of A and the consequent uniform continuity off .2Theorem 3. Let f : X → Y be a function. If f is continuous, then for any subset A ⊆ Xsatisfying∀x, y ∈ ∗A∃u ∈ Px,y with un ∈ ns(∗X) and st(un) ∈ σA, for all n (1)the following condition is valid∀z, w ∈ ∗f(A)∃v ∈ Pz,w with vn ∈ ns(∗Y ) and st(vn) ∈ σf(A), for all n.Proof. Let A be a set that satisfies condition (1). Given z and w in ∗f(A), let z = f(x) andw = f(y), for some x, y ∈ ∗A. Then, there exists u = (un)n=1,...,N ∈ Px,y, such that un ∈ ns(∗X)and st(un) ∈ σA, for all n = 1, . . . , N . Define vn := f(un), for all n = 1, . . . , N . It is easy to seethat v = (vn) satisfies the necessary conditions.Theorem 4. The set A is connected if∀x, y ∈ σA∃u ∈ Px,y with un ∈ ns(∗X) and st(un) ∈ σA, for all n. (2)Proof. Assume that A is not connected. Then A has a subset B /∈ {∅, A} that is simultaneouslyrelatively open and closed. Pick x ∈ σB, y ∈ σ(A−B) and u = (un)n=1,...,N ∈ Px,y such thatun ∈ ns(∗X) and st(un) ∈ σA, for all n, and define the internal setK := {n ∈ {1, . . . , N} |un ∈ ∗B}.Since K is nonempty (for example, 1 ∈ K), it has a maximum. Let k := maxK. Since y /∈ ∗Bthen k 6= N . Besides this, uk ∈ ∗B and uk+1 ∈ ∗(A−B). Since B and A − B are both closed,st(uk) ∈ B and st(uk+1) ∈ A−B.Since uk ≈ uk+1, the point st(uk) = st(uk+1) ∈ σB ∩ σ(A−B), which ends the proof.The previous condition is not enough to assert that A is path connected; e.g. take the set{(x, sin(1/x)) |x > 0} ∪ ({0} × [−1, 1]).However, if A is path connected then condition (2) is satisfied. Indeed, if we fix x, y ∈ σA, thenby hypothesis there exists a continuous path α : [0, 1]→ A with α(0) = x and α(1) = y. Take aninfinite N ∈ ∗N and define un := α( nN ) for n ∈ {0, . . . , N}. It is easy to check that (un) satisfiescondition (2).The converse of Theorem 4 is false in general, however we will obtain a related result.Theorem 5. If A is a connected set then for all x, y ∈ ∗A the condition Px,y 6= ∅ holds.Proof. Fix x, y ∈ σA and  ∈ σR+. Then there exists an -chain that joins x and y (c.f. [7], pag120). Therefore∀x, y ∈ σA∀ ∈ σR+ ∃N ∈ σN ∃{u2, . . . , uN−1} ⊂ σA∀i ∈ {1, . . . , N − 1} d(ui, ui+1) < ,where u1 := x and uN := y. Now, pick two points x, y ∈ ∗A. By the Transfer Principle, conditionholds with  ≈ 0.Observe that we actually proved that, for all infinitesimal , there exists u ∈ Px,y satisfyingd(ui, ui+1) < .Unfortunately, the d.i.p. need not to be nearstandard in A, as is shown in the next example.Let A be the subset of R2 defined by([0, 1]× {0}) ∪{(1n, y)|n ∈ N, y ∈ [0, 1]}∪ {(0, 0), (0, 1)} .The set A is connected but there is no d.i.p. joining the points (0, 0) to (0, 1) nearstandard inthe set.3Corollary 1. Let A be a compact set. Then A is connected if and only if∀x, y ∈ σA ∃u ∈ Px,y such that un ∈ ns(∗X) and st(un) ∈ σA, for all n.Proof. Follows from Theorems 4 and 5 and the fact that, for the nonstandard extension of compactsets, all points are nearstandard on the set.In conclusion, we have now a nice characterization of connected compact sets.Corollary 2. Let A be a non-empty set. Then A is connected and compact if and only if∀x, y ∈ ∗A∃u ∈ Px,y such that un ∈ ns(∗X) and st(un) ∈ σA, for all n. (3)Proof. We only need to prove that condition (3) implies the compactness condition. Fix x ∈ ∗A.By condition (3), there exists some u ∈ Px,x nearstandard on A. So u1 = x ∈ ns(∗X) andst(x) ∈ σA.AcknowledgmentsWork supported by Centre for Research on Optimization and Control (CEOC) from the “Fundac¸a˜opara a Cieˆncia e a Tecnologia” (FCT), cofinanced by the European Community Fund FEDER/POCI2010.References[1] R. Almeida, On the continuity of functions, Appl. Sci. (9), pp 1–4 (2007).[2] R. Almeida and D.F.M. Torres, Relaxed optimality conditions for mu-differentiable functions,Int. J. Appl. Math. & Stat. (14), No. M09 pp 53–66 (2009).[3] A.E. Hurd and P.A. Loeb, An Introduction to Nonstandard Real Analysis, Pure and AppliedMathematics, 118. Orlando etc., Academic Press, Inc. (1995).[4] S.C. Leth, Some Nonstandard Methods in Geometric Topology, in N.J. Cutland, V. Neves,F. Oliveira and J. Sousa-Pinto (Eds.), Developments in Nonstandard Mathematics, PitmanResearch Notes in Mathematics Series. 336, New York (1995).[5] S. Rodrigues, Propriedades da Separac¸a˜o, Conexa˜o e Real-compacidade, Master’s thesis, Uni-versity of Aveiro (2001).[6] A. Robinson, Non-Standard Analysis, North- Holland Publishing Company, Amsterdam(1974).[7] W. Sierpinski, General topology, (translated by C. Krieger), Dover Publications, Inc., Mineola,New York, (2000).[8] K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of Infinitesimals, PureAppl. Math. 72 New York-San Francisco-London, Academic Press (1976).4

Connectedness and compactness on standard sets

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