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)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9787c00-cfcc-43da-9235-b2564f9dd12e%2Faf44b3f3-e42c-4b11-b329-3c21ac596e8c%2Fg31i4tn_processed.jpeg&w=3840&q=75)
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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