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y nonempty subset is ercises 1. Let X be the space of real numbers with the usual topology, and let N be the integers. Find the derived set, the closure, the interior, and the boundary of N. 2. Let X be the set of reals, and let T = {SCX:0EX-S}U {X}. Show (1.2)

y nonempty subset is ercises 1. Let X be the space of real numbers with the usual topology, and let N be the integers. Find the derived set, the closure, the interior, and the boundary of N. 2. Let X be the set of reals, and let T = {SCX:0EX-S}U {X}. Show (1.2)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 34EQ
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Basic Topological Spaces: Please solve question 1 on paper. Please note it has parts to the question

(e) Let X be the space of real let A = (0, 1]. Then every subset of X is both ope A = cl(A) = int(A), and Fr(A) = 0. (d) Let X be the complex plane with the usual topology, and let S = {2:|2|<1}, and B = {z=(x,0): x ≥ 1}. Then cl(S) = {2: [2] ≤ 1}UB, int (S) = A, and Fr(S) = {z :|2|=1}UB. AUB, where A (e) Let X be the space of real numbers with the usual topology, and let A be the rational numbers. Then cl(A) = X, int(A) = 0), and Fr(A) = X. Definition A subset A of a topological space X is said to be dense if cl(A) = X. Thus the set of rational numbers is a dense subset of the space of real mbers with the usual topology, and in a space with the trivial topology, ery nonempty subset is dense. ercises 1. Let X be the space of real numbers with the usual topology, and let N be the integers. Find the derived set, the closure, the interior, and the boundary of N. 2. Let X be the set of reals, and let T = {SCX:0€ X-S} U{X}. Show that T is a topology for X and find the closure of the interval A = (1,2) and of the interval B = (-1,1). = 3. Let X be the set of positive integers. For each n E X, let Sn {k € X: k2n}. Show that T='{S: ne X} U {0} is a topology for X, and find the closure of the set of even integers. Find the closure of the singleton set A = {100}. 4. Let X be the space of reals with the cofinite topology (Example 2.1(d)), and let A be the positive integers and B = {1,2}. Find the derived set, the closure, the interior, and the boundary of each of the sets A and B.
Transcribed Image Text:(e) Let X be the space of real let A = (0, 1]. Then every subset of X is both ope A = cl(A) = int(A), and Fr(A) = 0. (d) Let X be the complex plane with the usual topology, and let S = {2:|2|<1}, and B = {z=(x,0): x ≥ 1}. Then cl(S) = {2: [2] ≤ 1}UB, int (S) = A, and Fr(S) = {z :|2|=1}UB. AUB, where A (e) Let X be the space of real numbers with the usual topology, and let A be the rational numbers. Then cl(A) = X, int(A) = 0), and Fr(A) = X. Definition A subset A of a topological space X is said to be dense if cl(A) = X. Thus the set of rational numbers is a dense subset of the space of real mbers with the usual topology, and in a space with the trivial topology, ery nonempty subset is dense. ercises 1. Let X be the space of real numbers with the usual topology, and let N be the integers. Find the derived set, the closure, the interior, and the boundary of N. 2. Let X be the set of reals, and let T = {SCX:0€ X-S} U{X}. Show that T is a topology for X and find the closure of the interval A = (1,2) and of the interval B = (-1,1). = 3. Let X be the set of positive integers. For each n E X, let Sn {k € X: k2n}. Show that T='{S: ne X} U {0} is a topology for X, and find the closure of the set of even integers. Find the closure of the singleton set A = {100}. 4. Let X be the space of reals with the cofinite topology (Example 2.1(d)), and let A be the positive integers and B = {1,2}. Find the derived set, the closure, the interior, and the boundary of each of the sets A and B.
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