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....
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...
. Let C be a collection of open subsets of R. Thus, C is a set whose elements are open subsets of R. Note that C need not be finite, or even countable. (a) Prove that the union U S is also an open subset of R. SEC (b) Assuming C is finite, prove that the intersection n S is an open subset of R. SEC (c) Give an example where C is infinite and n S is not open....
1. Let (X, d) be a metric space, and U, V, W CX subsets of X. (a) (i) Define what it means for U to be open. (ii) Define what it means for V to be closed. (iii) Define what it means for W to be compact. (b) Prove that in a metric space a compact subset is closed.
(1) Let X = {0}U[2,3], and give X the topology Tx = {0,{0}, [2, 3], X}. (a) (10 points) Is X To? Briefly justify your answer. (b) (10 points) Is X Hausdorff? Briefly justify your answer. (c) (10 points) Is X Tz? Briefly justify your answer. (d) (10 points) Is D = ({0} Ⓡ {0}) U ([2, 3] x [2, 3]) a closed subset of X x X with the product topology? Briefly justify your answer.
(a) Let (X, d) be a metric space. Prove that the complement of any finite set F C X is open. Note: The empty set is open. (b) Let X be a set containing infinitely many elements, and let d be a metric on X. Prove that X contains an open set U such that U and its complement UC = X\U are both infinite.
Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection between A and some infinite subset of N.) Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
3. (15 pts) Let D be an infinite set with cardinal d. Let A = {X C D | 0(X) <3}. Prove that o(A) = d.
1.5.7 Prove the following separately Theorem 1.5.10. Let (X,d) be a metric space. (a) IfY is a compact subset of X, and Z C Y, then Z is compact if and only if Z is closed (b) IfY. Y are a finite collection of compact subsets of X, then their union Y1 U...UYn is also compact. (c) Every finite subset of X (including the empty set) is compact.