Dynamical Systems Whose Orbit Spaces
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3. Closed Sets, Closures, and Density
3. Closed sets, closures, and density 1 Motivation Up to this point, all we have done is define what topologies are, define a way of comparing two topologies, define a method for more easily specifying a topology (as a collection of sets generated by a basis), and investigated some simple properties of bases. At this point, we will start introducing some more interesting definitions and phenomena one might encounter in a topological space, starting with the notions of closed sets and closures. Thinking back to some of the motivational concepts from the first lecture, this section will start us on the road to exploring what it means for two sets to be \close" to one another, or what it means for a point to be \close" to a set. We will draw heavily on our intuition about n convergent sequences in R when discussing the basic definitions in this section, and so we begin by recalling that definition from calculus/analysis. 1 n Definition 1.1. A sequence fxngn=1 is said to converge to a point x 2 R if for every > 0 there is a number N 2 N such that xn 2 B(x) for all n > N. 1 Remark 1.2. It is common to refer to the portion of a sequence fxngn=1 after some index 1 N|that is, the sequence fxngn=N+1|as a tail of the sequence. In this language, one would phrase the above definition as \for every > 0 there is a tail of the sequence inside B(x)." n Given what we have established about the topological space Rusual and its standard basis of -balls, we can see that this is equivalent to saying that there is a tail of the sequence inside any open set containing x; this is because the collection of -balls forms a basis for the usual topology, and thus given any open set U containing x there is an such that x 2 B(x) ⊆ U. -
What Are Lyapunov Exponents, and Why Are They Interesting?
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 1, January 2017, Pages 79–105 http://dx.doi.org/10.1090/bull/1552 Article electronically published on September 6, 2016 WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING? AMIE WILKINSON Introduction At the 2014 International Congress of Mathematicians in Seoul, South Korea, Franco-Brazilian mathematician Artur Avila was awarded the Fields Medal for “his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle.”1 Although it is not explicitly mentioned in this citation, there is a second unify- ing concept in Avila’s work that is closely tied with renormalization: Lyapunov (or characteristic) exponents. Lyapunov exponents play a key role in three areas of Avila’s research: smooth ergodic theory, billiards and translation surfaces, and the spectral theory of 1-dimensional Schr¨odinger operators. Here we take the op- portunity to explore these areas and reveal some underlying themes connecting exponents, chaotic dynamics and renormalization. But first, what are Lyapunov exponents? Let’s begin by viewing them in one of their natural habitats: the iterated barycentric subdivision of a triangle. When the midpoint of each side of a triangle is connected to its opposite vertex by a line segment, the three resulting segments meet in a point in the interior of the triangle. The barycentric subdivision of a triangle is the collection of 6 smaller triangles determined by these segments and the edges of the original triangle: Figure 1. Barycentric subdivision. Received by the editors August 2, 2016. -
The Continuity of Additive and Convex Functions, Which Are Upper Bounded
THE CONTINUITY OF ADDITIVE AND CONVEX FUNCTIONS, WHICH ARE UPPER BOUNDED ON NON-FLAT CONTINUA IN Rn TARAS BANAKH, ELIZA JABLO NSKA,´ WOJCIECH JABLO NSKI´ Abstract. We prove that for a continuum K ⊂ Rn the sum K+n of n copies of K has non-empty interior in Rn if and only if K is not flat in the sense that the affine hull of K coincides with Rn. Moreover, if K is locally connected and each non-empty open subset of K is not flat, then for any (analytic) non-meager subset A ⊂ K the sum A+n of n copies of A is not meager in Rn (and then the sum A+2n of 2n copies of the analytic set A has non-empty interior in Rn and the set (A − A)+n is a neighborhood of zero in Rn). This implies that a mid-convex function f : D → R, defined on an open convex subset D ⊂ Rn is continuous if it is upper bounded on some non-flat continuum in D or on a non-meager analytic subset of a locally connected nowhere flat subset of D. Let X be a linear topological space over the field of real numbers. A function f : X → R is called additive if f(x + y)= f(x)+ f(y) for all x, y ∈ X. R x+y f(x)+f(y) A function f : D → defined on a convex subset D ⊂ X is called mid-convex if f 2 ≤ 2 for all x, y ∈ D. Many classical results concerning additive or mid-convex functions state that the boundedness of such functions on “sufficiently large” sets implies their continuity. -
Math 501. Homework 2 September 15, 2009 ¶ 1. Prove Or
Math 501. Homework 2 September 15, 2009 ¶ 1. Prove or provide a counterexample: (a) If A ⊂ B, then A ⊂ B. (b) A ∪ B = A ∪ B (c) A ∩ B = A ∩ B S S (d) i∈I Ai = i∈I Ai T T (e) i∈I Ai = i∈I Ai Solution. (a) True. (b) True. Let x be in A ∪ B. Then x is in A or in B, or in both. If x is in A, then B(x, r) ∩ A , ∅ for all r > 0, and so B(x, r) ∩ (A ∪ B) , ∅ for all r > 0; that is, x is in A ∪ B. For the reverse containment, note that A ∪ B is a closed set that contains A ∪ B, there fore, it must contain the closure of A ∪ B. (c) False. Take A = (0, 1), B = (1, 2) in R. (d) False. Take An = [1/n, 1 − 1/n], n = 2, 3, ··· , in R. (e) False. ¶ 2. Let Y be a subset of a metric space X. Prove that for any subset S of Y, the closure of S in Y coincides with S ∩ Y, where S is the closure of S in X. ¶ 3. (a) If x1, x2, x3, ··· and y1, y2, y3, ··· both converge to x, then the sequence x1, y1, x2, y2, x3, y3, ··· also converges to x. (b) If x1, x2, x3, ··· converges to x and σ : N → N is a bijection, then xσ(1), xσ(2), xσ(3), ··· also converges to x. (c) If {xn} is a sequence that does not converge to y, then there is an open ball B(y, r) and a subsequence of {xn} outside B(y, r). -
Soft Somewhere Dense Sets on Soft Topological Spaces
Commun. Korean Math. Soc. 33 (2018), No. 4, pp. 1341{1356 https://doi.org/10.4134/CKMS.c170378 pISSN: 1225-1763 / eISSN: 2234-3024 SOFT SOMEWHERE DENSE SETS ON SOFT TOPOLOGICAL SPACES Tareq M. Al-shami Abstract. The author devotes this paper to defining a new class of generalized soft open sets, namely soft somewhere dense sets and to in- vestigating its main features. With the help of examples, we illustrate the relationships between soft somewhere dense sets and some celebrated generalizations of soft open sets, and point out that the soft somewhere dense subsets of a soft hyperconnected space coincide with the non-null soft β-open sets. Also, we give an equivalent condition for the soft cs- dense sets and verify that every soft set is soft somewhere dense or soft cs-dense. We show that a collection of all soft somewhere dense subsets of a strongly soft hyperconnected space forms a soft filter on the universe set, and this collection with a non-null soft set form a soft topology on the universe set as well. Moreover, we derive some important results such as the property of being a soft somewhere dense set is a soft topological property and the finite product of soft somewhere dense sets is soft some- where dense. In the end, we point out that the number of soft somewhere dense subsets of infinite soft topological space is infinite, and we present some results which associate soft somewhere dense sets with some soft topological concepts such as soft compact spaces and soft subspaces. -
DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V . -
Baire Category Theorem
Baire Category Theorem Alana Liteanu June 2, 2014 Abstract The notion of category stems from countability. The subsets of metric spaces are divided into two categories: first category and second category. Subsets of the first category can be thought of as small, and subsets of category two could be thought of as large, since it is usual that asset of the first category is a subset of some second category set; the verse inclusion never holds. Recall that a metric space is defined as a set with a distance function. Because this is the sole requirement on the set, the notion of category is versatile, and can be applied to various metric spaces, as is observed in Euclidian spaces, function spaces and sequence spaces. However, the Baire category theorem is used as a method of proving existence [1]. Contents 1 Definitions 1 2 A Proof of the Baire Category Theorem 3 3 The Versatility of the Baire Category Theorem 5 4 The Baire Category Theorem in the Metric Space 10 5 References 11 1 Definitions Definition 1.1: Limit Point.If A is a subset of X, then x 2 X is a limit point of X if each neighborhood of x contains a point of A distinct from x. [6] Definition 1.2: Dense Set. As with metric spaces, a subset D of a topological space X is dense in A if A ⊂ D¯. D is dense in A. A set D is dense if and only if there is some 1 point of D in each nonempty open set of X. -
1.1.3 Reminder of Some Simple Topological Concepts Definition 1.1.17
1. Preliminaries The Hausdorffcriterion could be paraphrased by saying that smaller neigh- borhoods make larger topologies. This is a very intuitive theorem, because the smaller the neighbourhoods are the easier it is for a set to contain neigh- bourhoods of all its points and so the more open sets there will be. Proof. Suppose τ τ . Fixed any point x X,letU (x). Then, since U ⇒ ⊆ ∈ ∈B is a neighbourhood of x in (X,τ), there exists O τ s.t. x O U.But ∈ ∈ ⊆ O τ implies by our assumption that O τ ,soU is also a neighbourhood ∈ ∈ of x in (X,τ ). Hence, by Definition 1.1.10 for (x), there exists V (x) B ∈B s.t. V U. ⊆ Conversely, let W τ. Then for each x W ,since (x) is a base of ⇐ ∈ ∈ B neighbourhoods w.r.t. τ,thereexistsU (x) such that x U W . Hence, ∈B ∈ ⊆ by assumption, there exists V (x)s.t.x V U W .ThenW τ . ∈B ∈ ⊆ ⊆ ∈ 1.1.3 Reminder of some simple topological concepts Definition 1.1.17. Given a topological space (X,τ) and a subset S of X,the subset or induced topology on S is defined by τ := S U U τ . That is, S { ∩ | ∈ } a subset of S is open in the subset topology if and only if it is the intersection of S with an open set in (X,τ). Alternatively, we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map ι : S X is continuous. -
Real Analysis, Math 821. Instructor: Dmitry Ryabogin Assignment
Real Analysis, Math 821. Instructor: Dmitry Ryabogin Assignment IV. 1. Problem 1. Definition 1. A set M ⊂ R is called closed if it coincides with its closure. In other words, M ⊂ R is closed, if it contains all of its limit points. Definition 2. A set A ⊂ R is called open if there is a closed set M ⊂ R such that A = R n M. In other words, A is open if all of its points are inner points, (a point x is called the inner point of a set A, if there exists > 0, such that the -neighbourhood U(x) of x, is contained in A, U(x) ⊂ A). Prove that any open set on the real line is a finite or infinite union of disjoint open intervals. (The sets (−∞; 1), (α, 1), (−∞; β) are intervals for us). Hint. Let G be open on R. Introduce in G the equivalence relationship, by saying that x ∼ y if there exists an interval (α, β), such that x; y 2 (α, β) ⊂ G. a) Prove at first that this is indeed the equivalence relationship, i. e., that x ∼ x, x ∼ y implies y ∼ x, and x ∼ y, y ∼ z imply x ∼ z. Conclude that G can be written as a disjoint union G = [τ2iIτ of \classes" of points Iτ , Iτ = fx 2 G : x ∼ τg, i is the set of different indices, i.e. τ 2 i if Iτ \ Ix = ?, 8x 2 G. b) Prove that each Iτ is an open interval. c) Show that there are at most countably many Iτ (the set i is countable). -
The Continuum Hypothesis and Its Relation to the Lusin Set
THE CONTINUUM HYPOTHESIS AND ITS RELATION TO THE LUSIN SET CLIVE CHANG Abstract. In this paper, we prove that the Continuum Hypothesis is equiv- alent to the existence of a subset of R called a Lusin set and the property that every subset of R with cardinality < c is of first category. Additionally, we note an interesting consequence of the measure of a Lusin set, specifically that it has measure zero. We introduce the concepts of ordinals and cardinals, as well as discuss some basic point-set topology. Contents 1. Introduction 1 2. Ordinals 2 3. Cardinals and Countability 2 4. The Continuum Hypothesis and Aleph Numbers 2 5. The Topology on R 3 6. Meagre (First Category) and Fσ Sets 3 7. The existence of a Lusin set, assuming CH 4 8. The Lebesque Measure on R 4 9. Additional Property of the Lusin set 4 10. Lemma: CH is equivalent to R being representable as an increasing chain of countable sets 5 11. The Converse: Proof of CH from the existence of a Lusin set and a property of R 6 12. Closing Comments 6 Acknowledgments 6 References 6 1. Introduction Throughout much of the early and middle twentieth century, the Continuum Hypothesis (CH) served as one of the premier problems in the foundations of math- ematical set theory, attracting the attention of countless famous mathematicians, most notably, G¨odel,Cantor, Cohen, and Hilbert. The conjecture first advanced by Cantor in 1877, makes a claim about the relationship between the cardinality of the continuum (R) and the cardinality of the natural numbers (N), in relation to infinite set hierarchy. -
Arxiv:1809.06453V3 [Math.CA]
NO FUNCTIONS CONTINUOUS ONLY AT POINTS IN A COUNTABLE DENSE SET CESAR E. SILVA AND YUXIN WU Abstract. We give a short proof that if a function is continuous on a count- able dense set, then it is continuous on an uncountable set. This is done for functions defined on nonempty complete metric spaces without isolated points, and the argument only uses that Cauchy sequences converge. We discuss how this theorem is a direct consequence of the Baire category theorem, and also discuss Volterra’s theorem and the history of this problem. We give a sim- ple example, for each complete metric space without isolated points and each countable subset, of a real-valued function that is discontinuous only on that subset. 1. Introduction. A function on the real numbers that is continuous only at 0 is given by f(x) = xD(x), where D(x) is Dirichlet’s function (i.e., the indicator function of the set of rational numbers). From this one can construct examples of functions continuous at only finitely many points, or only at integer points. It is also possible to construct a function that is continuous only on the set of irrationals; a well-known example is Thomae’s function, also called the generalized Dirichlet function—and see also the example at the end. The question arrises whether there is a function defined on the real numbers that is continuous only on the set of ra- tional numbers (i.e., continuous at each rational number and discontinuous at each irrational), and the answer has long been known to be no. -
Dense Sets on Bigeneralized Topological Spaces
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 21, 999 - 1003 HIKARI Ltd, www.m-hikari.com Dense Sets on Bigeneralized Topological Spaces Supunnee Sompong Department of Mathematics and Statistics Faculty of Science and Technology Sakon Nakhon Rajabhat University Sakon Nakhon 47000, Thailand s [email protected] Copyright c 2013 Supunnee Sompong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The purpose of this paper is to introduce the concept and some fundamental properties of dense sets on bigeneralized topological spaces. Keywords: bigeneralized topological spaces, dense sets 1 Introduction In 2010, the notion of bigeneralized topological spaces and the concepts of weakly functions were introduced by C. Boonpok [1]. He also investigated some of their characterizations. In 2012, S. Sompong and B. Rodjanadid [2], introduced a definition of dense sets in biminimal structure spaces and study some fundamental of their properties. In 2013, S. Sompong and S. Muangchan [3, 4], introduced the definition of boundary and exterior set on bigeneralized topological spaces and the concepts of basic properties. In this paper, we introduce the concept of dense sets on bigeneralized topo- logical spaces and study some properties. 2 Preliminaries Let X be a nonempty set and g be a collection of subsets of X. Then g is called a generalized topology (briefly GT)onX if ∅∈g and if Gi ∈ g for i ∈ I = ∅ then G = i∈I Gi ∈ g. 1000 S. Sompong By (X, g), we denote a nonempty set X with a generalized topology g on X and it is called a generalized topological space (briefly GTS) on X.