THE REAL LINE

  1.3 THE REAL LINE

 One of our objectives is to develop rigorously the concepts of limit, continuity, differentiability, and integrability, which you have seen in calculus. To do this requires a better understanding of the real numbers than is provided in calculus. The purpose of this section is to develop this understanding. Since the utility of the concepts introduced here will not become apparent until we are well into the study of limits and continuity, you should reserve judgment on their value until they are applied. As this occurs, you should reread the applicable parts of this section. This applies especially to the concept of an open covering and to the Heine–Borel and Bolzano–Weierstrass theorems, which will seem mysterious at first. We assume that you are familiar with the geometric interpretation of the real numbers as points on a line. We will not prove that this interpretation is legitimate, for two reasons: (1) the proof requires an excursion into the foundations of Euclidean geometry, which is not the purpose of this book; (2) although we will use geometric terminology and intuition in discussing the reals, we will base all proofs on properties (A)–(I) (Section 1.1) and their consequences, not on geometric arguments. Henceforth, we will use the terms real number system and real line synonymously and denote both by the symbol R; also, we will often refer to a real number as a point (on the real line). Some Set Theory In this section we are interested in sets of points on the real line; however, we will consider other kinds of sets in subsequent sections. The following definition applies to arbitrary sets, with the understanding that the members of all sets under consideration in any given context come from a specific collection of elements, called the universal set. In this section the universal set is the real numbers. Definition 1.3.1 Let S and T be sets. (a) S contains T , and we write S  T or T  S, if every member of T is also in S. In this case, T is a subset of S. (b) S T is the set of elements that are in S but not in T . (c) S equals T , and we write S D T , if S contains T and T contains S; thus, S D T if and only if S and T have the same member 20 Chapter 1 The Real Numbers (d) S strictly contains T if S contains T but T does not contain S; that is, if every member of T is also in S, but at least one member of S is not in T (Figure 1.3.1). (e) The complement of S, denoted by S c , is the set of elements in the universal set that are not in S. (f) The union of S and T , denoted by S [ T , is the set of elements in at least one of S and T (Figure 1.3.1(b)). (g) The intersection of S and T , denoted by S \ T , is the set of elements in both S and T (Figure 1.3.1(c)). If S \ T D ; (the empty set), then S and T are disjoint sets (Figure 1.3.1(d)). (h) A set with only one member x0 is a singleton set, denoted by fx0g.

Comments

Post a Comment

Popular posts from this blog

Functions between metric spaces

Image
  Functions between metric spaces [ edit ] Euler diagram  of types of functions between metric spaces. Unlike in the case of topological spaces or algebraic structures such as  groups  or  rings , there is no single "right" type of  structure-preserving function  between metric spaces. Instead, one works with different types of functions depending on one's goals. Throughout this section, suppose that  ( � 1 , � 1 )  and  ( � 2 , � 2 )  are two metric spaces. The words "function" and "map" are used interchangeably. Isometries [ edit ] Main article:  Isometry One interpretation of a "structure-preserving" map is one that fully preserves the distance function: A function  � : � 1 → � 2  is  distance-preserving [12]  if for every pair of points  x  and  y  in  M 1 , � 2 ( � ( � ) , � ( � ) ) = � 1 ( � , � ) . It follows from the metric space axioms that a distance-preserving function is injective. A bijective distance-preserving function is calle
Read more

Quasi-isometries

Image
  Quasi-isometries [ edit ] Main article:  Quasi-isometry A  quasi-isometry  is a map that preserves the "large-scale structure" of a metric space. Quasi-isometries need not be continuous. For example,  � 2  and its subspace  � 2  are quasi-isometric, even though one is connected and the other is discrete. The equivalence relation of quasi-isometry is important in  geometric group theory : the  Švarc–Milnor lemma  states that all spaces on which a group  acts geometrically  are quasi-isometric. [15] Formally, the map  � : � 1 → � 2  is a  quasi-isometric embedding  if there exist constants  A  ≥ 1  and  B  ≥ 0  such that 1 � � 2 ( � ( � ) , � ( � ) ) − � ≤ � 1 ( � , � ) ≤ � � 2 ( � ( � ) , � ( � ) ) + �  for all  � , � ∈ � 1 . It is a  quasi-isometry  if in addition it is  quasi-surjective , i.e. there is a constant  C  ≥ 0  such that every point in  � 2  is at distance at most  C  from some point in the image  � ( � 1 ) . Notions of metric space equivalence [ edit ] See also
Read more