Prove: Let A be a dense subset of (X, T), and let B be a non-empty...
Let X be a non-empty set. Show that the only dense subset of X with respect to the discrete metric ddise is X. The whole set of any metric spaces is always dense, so this question is really asking you to exclude all other possibilities. Show that if (X, d) is a metric space and has dense subset A + X, then (X, d) is not topologically equivalent to (X, ddisc). (Note that this is another way of showing that...
Let A be a non empty subset of R that is bounded below and let a=inf A. If a a&A, prove thal x is a limit point of A
Let A be a non-empty subset of R that is bounded above. (a) Let U = {x ∈ R : x is an upper bound for A}, the set of all upper bounds for A. Prove that there exists a u ∈ R such that U = [u, ∞). (b) Prove that for all ε > 0 there exists an x ∈ A such that u − ε < x ≤ u. This u is one shown to exist in...
6. Let X be a non-empty subset of an ordered field with the least upper bound property. Supposed that X is bounded above and define -X = {-1 : TEX} Prove that supX = - inf(-X).
Let X be a set and let T be the family of subsets U of X such that X\U (the complement of U) is at most countable, together with the empty set. a) Prove that T is a topology for X. b) Describe the convergent sequences in X with respect to this topology. Prove that if X is uncountable, then there is a subset S of X whose closure contains points that are not limits of the sequences in S....
4·Let A and B be non-empty subsets of a space X. Prove that A U B is disconnected if A n B)U(A nB) 0. Prove that X is connected if and only if for every pair of non-empty subsets A and B of X such that X A U B we have (A B)U (An B)O.
9. Let X and Y be metric spaces, and let D be a dense subset of X. (For the definition of "dense, see Problem 4 at the end of Section 3.5.) (a) Let f : X → Y and g : X → Y be continuous functions. Suppose that f(d)gld) for all d E D. Prove that f and g are the same function.
Exercise 6.22 Given a non-empty subset A of a metric space X, prove that a pointe X is in a A iff da, A) = 0 = día, X\A), where daz, A) is defined as in Exercise 6.16.
4. Let A, B CR be non-empty open sets. Prove that AU B is an open set.
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...