DOCSLIB.ORG

Existence of a Limit on a Dense Set, and Construction of Continuous Functions on Special Sets

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Existence of a Limit on a Dense Set, and Construction of Continuous Functions on Special Sets Existence of a Limit on a Dense Set, and Construction of Continuous Functions on Special Sets Goodi Khalsa, Charles Ouyang, and Hannah Skavdal REU 2012 Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Recap: Definitions Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition Given a real-valued function f , the limit of f exists at a point c 2 R if for each given " > 0, there exists δ > 0 such that for any x 2 R if 0 < jc − xj < δ, then jf (c) − Lj < ". Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition Given a real-valued function f , f is continuous at a point c 2 R if for each given " > 0, there exists δ > 0 such that for any x 2 R if jc − xj < δ, then jf (c) − f (x)j < ". Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Notation Given f :[a; b] ! R, let FC denote the set of points where f is continuous. Let F+ denote the set of points where the right-sided limit of f exists. Similarly, let F− denote the set of points where the left-sided limit of f exists. Then, also let FL designate the set of points where a one-sided limit exists, that is FL = F+ [ F−. Definition Let B be a subset of R. B is said to be dense in R if for any point x 2 R, x is either in B or a limit point of B. Equivalently, given any " > 0, and any x 2 R, then 9p 2 B such that p 2 N"(x). Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition Let B be a subset of R. B is said to be dense in R if for any point x 2 R, x is either in B or a limit point of B. Equivalently, given any " > 0, and any x 2 R, then 9p 2 B such that p 2 N"(x). Notation Given f :[a; b] ! R, let FC denote the set of points where f is continuous. Let F+ denote the set of points where the right-sided limit of f exists. Similarly, let F− denote the set of points where the left-sided limit of f exists. Then, also let FL designate the set of points where a one-sided limit exists, that is FL = F+ [ F−. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Notation Let D be the set of points of discontinuity of a real function f . Then each point in D belongs to one of the following 3 sets: D1, the set of points c 2 [a; b] such that f has either a removable or jump discontinuity at c. D2, the set of points c 2 [a; b] such that f has only either a right-sided limit or a left-sided limit at c. D3, the set of points c 2 [a; b] such that f has neither a right-sided limit or a left-sided limit at c. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition A set A is countable if there exists a one-to-one correspondence from A to N. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Recap: Results Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Proposition Let f be a function defined on [a; b]. Then the sets D1 and D2 are at most countable. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Theorem Let f be a function defined on [a; b]. Assume that the set FL is dense in [a; b]. Then the set FC is nonempty, dense in [a; b], and uncountable. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Theorem Let f :[a; b] ! R, with FL dense in R. Then f is unique on FC . Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Corollary If f :[a; b] ! R and m(D3) = 0, and given FL dense in R, then f is unique except at a set of measure zero. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Proposition Given a function f :[a; b] ! R, there is no countable dense set G where G := fc : lim f (x) = L with L 2 g. x!c R Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets The set of continuity points of a function f : R ! R is a Gδ set. Conversely, every Gδ subset of R is the set of continuity points of a function f : R ! R. Definition An Fσ set is a countable union of closed sets. A Gδ set is a countable intersection of open sets. Examples: Q is Fσ. RnQ is Gδ. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition An Fσ set is a countable union of closed sets. A Gδ set is a countable intersection of open sets. Examples: Q is Fσ. RnQ is Gδ. The set of continuity points of a function f : R ! R is a Gδ set. Conversely, every Gδ subset of R is the set of continuity points of a function f : R ! R. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Let !(f ; J) = sup f (x) − inf f (x) x2J x2J !(f ; a) = inf (!(f ; J)) a2J 1 Er = fa : !(f ; a) ≥ r g Theorem c (FL) is an Fσ set. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets !(f ; a) = inf (!(f ; J)) a2J 1 Er = fa : !(f ; a) ≥ r g Theorem c (FL) is an Fσ set. Let !(f ; J) = sup f (x) − inf f (x) x2J x2J Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets 1 Er = fa : !(f ; a) ≥ r g Theorem c (FL) is an Fσ set. Let !(f ; J) = sup f (x) − inf f (x) x2J x2J !(f ; a) = inf (!(f ; J)) a2J Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Theorem c (FL) is an Fσ set. Let !(f ; J) = sup f (x) − inf f (x) x2J x2J !(f ; a) = inf (!(f ; J)) a2J 1 Er = fa : !(f ; a) ≥ r g Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Theorem There is no function that is continuous only on Q. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Is there a Gδ set of measure zero, containing Q, that we can construct a continuous function on? Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition A real number x is a Liouville number if for any given n 2 N, there exist infinitely many relatively prime integers p and q with q > 1 p 1 such that 0 < jx − q j < qn . We will denote the set of Liouville numbers with L. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Let f : R ! R 0; if x 2 L [ f (x) = Q 1 ; otherwise Nx Where Nx is defined as the first n where x fails to meet the definition of a Liouville number. In other words, the first n such that there do not exist p and q such that p 1 0 < jx − q j < qn . Is there a Gδ set of measure zero, containing Q, that we can construct a continuous function on? Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Is there a Gδ set of measure zero, containing Q, that we can construct a continuous function on? Let f : R ! R 0; if x 2 L [ f (x) = Q 1 ; otherwise Nx Where Nx is defined as the first n where x fails to meet the definition of a Liouville number. In other words, the first n such that there do not exist p and q such that p 1 0 < jx − q j < qn . Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition The irrationality measure for a real number x is a numeric representation of how well x can be approximated by the rationals. p 1 Let µ be the least upper bound such that 0 < jx − q j < qµ where p; q 2 Z. We call µ the irrationality measure of x. Notation Let µ(x) stand for irrationality measure of x. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition If r is a root of a nonzero polynomial equation n n−1 anx + an−1x + ··· + a1x + a0 = 0 where ai 2 Z and r satisfies no similar equation of degree < n then r is said to be an algebraic number of degree n. A number that is not algebraic is said to be transcendental. For example: µ(x) = 1 if x 2 L, µ(x) = 1 if x 2 Q, µ(x) = 2 if x is an algebraic number of degree > 1, µ(x) ≥ 2 if x is transcendental. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets For example: µ(x) = 1 if x 2 L, µ(x) = 1 if x 2 Q, µ(x) = 2 if x is an algebraic number of degree > 1, µ(x) ≥ 2 if x is transcendental.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Topological Vector Spaces
    Topological Vector Spaces Maria Infusino University of Konstanz Winter Semester 2015/2016 Contents 1 Preliminaries3 1.1 Topological spaces ......................... 3 1.1.1 The notion of topological space.............. 3 1.1.2 Comparison of topologies ................. 6 1.1.3 Reminder of some simple topological concepts...... 8 1.1.4 Mappings between topological spaces........... 11 1.1.5 Hausdorff spaces...................... 13 1.2 Linear mappings between vector spaces ............. 14 2 Topological Vector Spaces 17 2.1 Definition and main properties of a topological vector space . 17 2.2 Hausdorff topological vector spaces................ 24 2.3 Quotient topological vector spaces ................ 25 2.4 Continuous linear mappings between t.v.s............. 29 2.5 Completeness for t.v.s........................ 31 1 3 Finite dimensional topological vector spaces 43 3.1 Finite dimensional Hausdorff t.v.s................. 43 3.2 Connection between local compactness and finite dimensionality 46 4 Locally convex topological vector spaces 49 4.1 Definition by neighbourhoods................... 49 4.2 Connection to seminorms ..................... 54 4.3 Hausdorff locally convex t.v.s................... 64 4.4 The finest locally convex topology ................ 67 4.5 Direct limit topology on a countable dimensional t.v.s. 69 4.6 Continuity of linear mappings on locally convex spaces . 71 5 The Hahn-Banach Theorem and its applications 73 5.1 The Hahn-Banach Theorem.................... 73 5.2 Applications of Hahn-Banach theorem.............. 77 5.2.1 Separation of convex subsets of a real t.v.s. 78 5.2.2 Multivariate real moment problem............ 80 Chapter 1 Preliminaries 1.1 Topological spaces 1.1.1 The notion of topological space The topology on a set X is usually defined by specifying its open subsets of X.
    [Show full text]
  • 7560 THEOREMS with PROOFS Definition a Compactification of A
    7560 THEOREMS WITH PROOFS De¯nition A compacti¯cation of a completely regular space X is a compact T2- space ®X and a homeomorphic embedding ® : X ! ®X such that ®(X) is dense in ®X. Remark. To explain the separation assumptions in the above de¯nition: we will only be concerned with compacti¯cations which are Hausdor®. Note that only completely regular spaces can have a T2-compacti¯cation. Remark. A few of the concepts and results below are from ¯rst year topology, and are included here for completeness. De¯nition. Let X be a locally compact non-compact Hausdor® space. The one-point compacti¯cation !X of X is the space X [ f1g, where points of X have their usual neighborhoods, and a neighborhood of 1 has the form f1g [ (X n K) where K is a compact subset of X. In the framework of the previous de¯nition, the map ! : X ! !X is the identity map on X. Of course, X is open in its one-point compacti¯cation. We'll see that this is the case for any compacti¯cation of a locally compact X. Lemma 1. If X is a dense subset of a T2 space Z, and X is locally compact, then X is open in Z. Proof. If x 2 X, let Nx be a compact neighborhood x in X and Ux = int(Nx). Then Ux = X \ Vx for some Vx open in Z. Thus Ux is dense in Vx and so Vx ½ V x = U x ½ N x = Nx where closures are taken in Z.
    [Show full text]
  • Topological Vector Spaces and Continuous Linear Functionals
    CHAPTER III TOPOLOGICAL VECTOR SPACES AND CONTINUOUS LINEAR FUNCTIONALS The marvelous interaction between linearity and topology is introduced in this chapter. Although the most familiar examples of this interaction may be normed linear spaces, we have in mind here the more subtle, and perhaps more important, topological vector spaces whose topologies are defined as the weakest topologies making certain collections of functions continuous. See the examples in Exercises 3.8 and 3.9, and particularly the Schwartz space S discussed in Exercise 3.10. DEFINITION. A topological vector space is a real (or complex) vector space X on which there is a Hausdorff topology such that: (1) The map (x, y) → x+y is continuous from X×X into X. (Addition is continuous.) and (2) The map (t, x) → tx is continuous from R × X into X (or C × X into X). (Scalar multiplication is continuous.) We say that a topological vector space X is a real or complex topolog- ical vector space according to which field of scalars we are considering. A complex topological vector space is obviously also a real topological vector space. A metric d on a vector space X is called translation-invariant if d(x+ z, y + z) = d(x, y) for all x, y, z ∈ X. If the topology on a topological vector space X is determined by a translation-invariant metric d, we call X (or (X, d)) a metrizable vector space. If x is an element of a metrizable vector space (X, d), we denote by B(x) the ball of radius 43 44 CHAPTER III around x; i.e., B(x) = {y : d(x, y) < }.
    [Show full text]
  • Planetmath: Topological Space
    (more info) Math for the people, by the people. Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS Advanced search topological space (Definition) "topological space" is owned by djao. [ full author list (2) ] (more info) Math for the people, by the people. Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS Advanced search compact (Definition) "compact" is owned by djao. [ full author list (2) ] Dense set 1 Dense set In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if any point x in X belongs to A or is a limit point of A.[1] Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A - for instance, every real number is either a rational number or has one arbitrarily close to it (see Diophantine approximation). Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one point from A. Equivalently, A is dense in X if and only if the only closed subset of X containing A is X itself. This can also be expressed by saying that the closure of A is X, or that the interior of the complement of A is empty. The density of a topological space X is the least cardinality of a dense subset of X. Density in metric spaces An alternative definition of dense set in the case of metric spaces is the following. When the topology of X is given by a metric, the closure of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points), Then A is dense in X if Note that .
    [Show full text]
  • Measure and Category
    Measure and Category Marianna Cs¨ornyei [email protected] http:/www.ucl.ac.uk/∼ucahmcs 1 / 96 A (very short) Introduction to Cardinals I The cardinality of a set A is equal to the cardinality of a set B, denoted |A| = |B|, if there exists a bijection from A to B. I A countable set A is an infinite set that has the same cardinality as the set of natural numbers N. That is, the elements of the set can be listed in a sequence A = {a1, a2, a3,... }. If an infinite set is not countable, we say it is uncountable. I The cardinality of the set of real numbers R is called continuum. 2 / 96 Examples of Countable Sets I The set of integers Z = {0, 1, −1, 2, −2, 3, −3,... } is countable. I The set of rationals Q is countable. For each positive integer k there are only a finite number of p rational numbers q in reduced form for which |p| + q = k. List those for which k = 1, then those for which k = 2, and so on: 0 1 −1 2 −2 1 −1 1 −1 = , , , , , , , , ,... Q 1 1 1 1 1 2 2 3 3 I Countable union of countable sets is countable. This follows from the fact that N can be decomposed as the union of countable many sequences: 1, 2, 4, 8, 16,... 3, 6, 12, 24,... 5, 10, 20, 40,... 7, 14, 28, 56,... 3 / 96 Cantor Theorem Theorem (Cantor) For any sequence of real numbers x1, x2, x3,..
    [Show full text]