topology Problem 1. (1) Suppose Ti and Tz are two different topologies on a set X. When is the identity map id X X given by id(r) (2) Show that the subspace topology Ty is the smallest topology on YcX for which the inclusion : Y+X is a continuous map. = ra continuous map from (X, Ti) to (X, T2)?
10. Let T1 and T2 be two topologies on a set X. Then T1 is said to be a finer topology than T2 (and T2 is said to be a coarser topology than T1) if Ti2 T2. Prove that (i) the Euclidean topology R is finer than the finite-closed topology on R; (ii) the identity function f: (X, Ti) -(X, T2) is continuous if and only if TI is a finer topology than T2.
Instruction: Do any 10 of the 14 questions. Each question is worth 10 points. (For each True/False question, if it is true, answer T and give reasons for your answer. If it is false, answer F and give a explicit counterexample or other explanation of why it is false.) 1) True or False: The set {x : x = : x = tany, y e [0,5)} is an compact subset of R, the set of all real numbers with the...
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....
1. [15 pts] Use Definition 1.5 (definition of probability function) to prove Propo- sition 1.3 () 15 pts) & (iv) [10 pts). You do not need to prove (i) and (ii). [Definition 1.5/ Let Ω be a set of all possible events. A probability function P : Ω → 0,11 satisfies the follouing three conditions (i) 0s P(A) S 1 for any event A; (iii) For any sequence of mutually exclusive events A1, A2 ,A", i.e. A, n Aj =...
(a) Let x(t) = 1 when 0 <t<1 and 0 for all other real t. Find and graph the following: (i) r(t -3). [5] (ii) c(t/2). (5] (iii) <((t-3)/2). [5] (iv) (t/2) – 3). [5]
3. (a) Let (R, τe) be the usual topology on R. Find the limit point set of the following subsets of R (i) A = { n+1 n : n ∈ N} (ii) B = (0, 1] (iii) C = {x : x ∈ (0, 1), x is a rational number (b) Let X denote the indiscrete topology. Find the limit point set A 0 of any subset A of X. (c) Prove that a subset D of X is...
Problem 5. For u = (Uk)x=1,2,... El, we set Tnu = (U1, U2, ..., Un, 0,...). (1) Prove that Tn E B(C2, (). (2) We define the operator I as Iu = u (u € 14). Then, prove that for any u ele, lim ||T,u - Tulee = 0. (3) Prove that I, does not converge to I with respect to the norm of B(C²,1). Let X, Y be Banach spaces. Definition (review) We denote by B(X, Y) a set...
Example: Let x, y ∈ Rn, where n ∈ N. The line segment joining x to y is the subset {(1 − t)x + ty : 0 ≤ t ≤ 1 } of R n . A subset A of Rn, where n ∈ N, is called convex if it contains the line segment joining any two of its points. It is easy to check that any convex set is path-connected. (a) Let f : X → Y be an...
Let X be the space of all continuous functions from [0, 1] to [0, 1] equipped with the sup metric. Let Xi be the set of injective and Xs be the set of surjective elements of A and let Xis = Xi ∩ Xs. Prove or disprove: i) Xi is closed, ii) Xs is closed, iii) Xis is closed, iv) X is connected, v) X is compact.