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CHAPTER 4 – BASIC PROPERTIES
METRIC SPACES TEST QUESTIONS
AND ANSWERS
4.1 - Basic Properties of Metric Spaces 4.1 - Basic Properties of Metric Spaces What does the real number form? A chain in which any set bounded from above has a least upper bound, and dually, any set bounded from below has a greater lower bound. A systematic notation for the least upper bound sup. If A is a set of real numbers bounded from above, we write sup A for the least upper bound of A. Notation for when A is not bounded from above sup A = ∞ Notation for greatest lower bound inf What can we use instead of sup and inf if A is finite? max and min instead of sup and inf Definition - Metric space A metric space is a set M equipped with a real-valued function D(a,b) defined for all a,b∈M so as to satisfy: I. D(a,a)=0 for all a. II. D(a,b)>0 for a≠b.
III. D(a,b)=D(b,a). IV. D(a,c) ≤ D(a,b) + D(b,c) (triangular inequality) How can we think about the function D(a,b)? As the distance between a and b. Example of a metric space The set of all real numbers, with D(a,b) = |a-b| Any set of real numbers is a metric space relative to the distance function D(a,b) = |a-b| Definition - The Euclidean plane The set of all ordered pairs of real numbers. Definition - Euclidean distance Let M be the usual Euclidean plane: the set of all ordered pairs of real numbers. Two typical points of M are u = (x₁,y₁), v = (x₂,y₂). The Euclidean distance is given by D(u,v) = √(x₁-x₂)²+(y₁-y₂)² Ath. að rótin nær yfir allt! An arbitrary set equipped with a trivial distance function. If M is any set, take D(a,a)=0 and D(a,b)=1 for a≠b in M. Theorem 26 - For any points a,b,c in a metric space we have |D(a,c) - D(b,c)|≤ D(a,b) Definition - Diameter The diameter of a metric M is sup(D(a,b)), taken over all a,b∈M. 4.2 - Open sets 4.2 - Open sets Definition- Isometry
A neighbourhood of a point x in a metric space is a subset containing an open set containing x. What is equivalent to requiring that the neighbourhood contains an open ball containing x that in cointains an open ball with center x. Example of a neighbourhood A closed ball with center x. Theorem 31 - A subset of a metric space is open if and only if it contains, along with any point x, some neighbourhood of x. Definition - Isolated point A point x in a metric space is called isolated if the set {x} consisting of x alone is open. 4.3 - Convergence; closed sets 4.3 - Convergence; closed sets Definition - Discrete metric space A metric space is called discrete if every subset is open. (By Theorem 28 it is equivalent to say that every point is isolated). Definition - Convergence of a sequence Let x₁,x₂,x₃,..., be a sequence in a metric space M and let x∈M. The sequence is said to converge to x if for any positive real number ε there exists an integer N (depending on ε) such that D(x_i,x)<ε for all i≥N. x is called the limit of the sequence. The elements of the sequence need not be distinct, and they need not differ from x. {x_i} A handy symbol for the sequence x₁,x₂,x₃,...,. What do we write for the statement that the sequence {x_i} converges to x?
x_i→x When does a sequence converge to x? If every neighbourhood of x contains the sequence after deletion of a finite initital segment. Definition - A closed set A closed set is one which contains all limits of its convergent sequences. Example of closed sets. M is a closed set. The empty set is a closed set. In fact, M and the empty set are both open and closed. Theorem 32 - Let A be any subset and x any point in a metric space. There exists a sequence of elements of A converging to x if and only if every S_r(x)∩A is nonempty. Theorem 33 - A subset of a metric space is closed if and only if its complement is open. Theorem 34 - The intersection of any collection of closed sets in a metric space is again a closed set. Theorem 35 - The union of a finite number of closed sets in a metric space is again a closed set. Definition - Closure of A Let A_ (A bar) denote the intersection of all closed subsets of M containing A. Then A_ is closed, and in an obvious sense it is the smallest closed set containing A. Theorem 36 - Let A be a subset and x a point in a metric space M. The following three statements are equivalent:
Theorem 40 - Let f be a mapping from a metric space X to a metrix space Y. Then the following three statements are equivalent: (a) f is continuous. (b) The complete inverse image of an open set is open, i.e. for any open set V in Y, f ^-1(V) is an open subset of X. (c) The complete inverse image of a closed set is closed, i.e., for any closed set G in Y, f^-1(G) is a closed subset of X. Definition - A dense subset A subset A of a metric space M is dense in M if A_=M. Theorem 41 - Let f and g be continuous functions from a metric space X to a metric space Y. Let B be the set of all x in X for which f(x)=g(x). Then B is a closed subset of X. Theorem 42 - Let A be a dense subset of a metric space X, and let f and g be continuous functions from X to a metric space Y. Suppose that f and g coincide (falla saman) on A. Then they coincide on all of X.
