let X = {0,1,2,3,4} and A = {...}.
a) Prove A is an albegra of subsets of X
b) Is A an ... albegra of subsets
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5. Let X be a topological space and let A and B be connected subsets of X. Prove that if AndB+, then AUB is connected.
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.
(1) Let X and Y be sets. Let f be a function from X to Y, (a) IF BEY, recall that F-'(B) = {xeX \flyeBX(y,x) ef-)}. Prove that f'(B)={xeX | fk)e B}. (hint: Reprember that even though t is a thought is a function, the relation f may well not be itself a function.) Al b) Let {B; \je J} be an inbred family of subsets of Y. Prove that of "b) = f'(21B;).
5. Let f: X → Y. Prove that for any indexed family (Ai);el of subsets of Y iEI iEI iEI iEI 5. Let f: X → Y. Prove that for any indexed family (Ai);el of subsets of Y iEI iEI iEI iEI
4. Consider a function f : X → Y. 4a) (5 pts) Let C, D be subsets of Y. Prove that f (CND)sf1(C)nf-1(D). 4b) (10 pts) Let A, B be subsets of X and assume the function f be one-to- one. Prove that f(A) n f(B)Cf(An B) (Justify each of your steps.) 4c) (4pts) Find an example showing that if the function f is not one-to-on the inequality (1) is violated.
Let M be a ơ-algebra of subsets of X and P the set of finite measures on M. Prove that (a) d(μ, v) = supAEM |μ( A)-v(A)| defines a metric on P (b) (P, d) is complete. Let M be a ơ-algebra of subsets of X and P the set of finite measures on M. Prove that (a) d(μ, v) = supAEM |μ( A)-v(A)| defines a metric on P (b) (P, d) is complete.
3. (a) (5 points) On the set A= R\{0}, let x ~ y if and only if x · y > 0. Is this relation an equivalence relation? Prove your answer. (b) (5 points) Let B = {1, 2, 3, 4, 5} and C = {1,3}. On the set of subsets of B, let D ~ E if and only if DAC = EnC. Is this relation an equivalence relation? Prove your answer.
ne [0,1,2,3,4}. Determine Sn = {X:X C {1,2,B},|X| = n}. Suppose
(c) Let f : IR2 -R2 be given by f(x,)= (a1)2-y1, (-12) Let S, S' be the subsets of R2 as indicated in the picture below. Prove that f maps S onto S' (0,1) v-axis V=1 (2,1) (1,1) y =(x-1)2 у-ахis u 1 v=u-1 u-axis (1,0) (0,0) х-аxis (1,0) (c) Let f : IR2 -R2 be given by f(x,)= (a1)2-y1, (-12) Let S, S' be the subsets of R2 as indicated in the picture below. Prove that f maps S...