Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
(e) Let X = [0, 1] U [2, 3] with the subspace topology of the real line. Then A = [0, 1] and B = [2, 3] are disjoint, non-empty open subsets of X for which X = A U B, so X is disconnected. (Note that A and B are relatively open since A = (-∞, 3/2) N X and B = (3/2, ∞0) n X.) (f) Let X denote the set of real nụmbers with the one additional point a = (0, 1), and let the topology T for X consist of Ø and all subsets of X which contain a. Then there do not exist two disjoint non-empty open sets, so X is connected. Note that as a subspace of (X, T ), R is assigned the discrete topology and is therefore disconnected. This example (as well as examples (c), (d), and (e) above) demonstrate that the property of being connected is definitely not hereditary. %3D

(e) Let X = [0, 1] U [2, 3] with the subspace topology of the real line. Then A = [0, 1] and B = [2, 3] are disjoint, non-empty open subsets of X for which X = A U B, so X is disconnected. (Note that A and B are relatively open since A = (-∞, 3/2) N X and B = (3/2, ∞0) n X.) (f) Let X denote the set of real nụmbers with the one additional point a = (0, 1), and let the topology T for X consist of Ø and all subsets of X which contain a. Then there do not exist two disjoint non-empty open sets, so X is connected. Note that as a subspace of (X, T ), R is assigned the discrete topology and is therefore disconnected. This example (as well as examples (c), (d), and (e) above) demonstrate that the property of being connected is definitely not hereditary. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Topic Video
Question
100%
Justify the examples (e) and (f) in details
Definition: A topological space X is disconnected or separated if it is the union of two disjoint, non-empty open sets. Such a pair A, B of subsets of X is called a separation of X. A space X is connected provided that it is not disconnected. In other words, X is connected if there do not exist open subsets A and B of X such that A + Ø, B+ Ø, ANB= Ø, AUB= X. %3D A subspace Y of X is connected provided that it is a connected space when assigned the subspace topology. The terms connected set and connected subset are sometimes used to mean connected space and connected subspace, respectively.
Transcribed Image Text:Definition: A topological space X is disconnected or separated if it is the union of two disjoint, non-empty open sets. Such a pair A, B of subsets of X is called a separation of X. A space X is connected provided that it is not disconnected. In other words, X is connected if there do not exist open subsets A and B of X such that A + Ø, B+ Ø, ANB= Ø, AUB= X. %3D A subspace Y of X is connected provided that it is a connected space when assigned the subspace topology. The terms connected set and connected subset are sometimes used to mean connected space and connected subspace, respectively.
(e) Let X = [0, 1] U [2, 3] with the subspace topology of the real line. Then A = [0, 1] and B = [2, 3] are disjoint, non-empty open subsets of X for which X = A U B, so X is disconnected. (Note that A and B are relatively open since A = (-∞, 3/2) n X and B = (3/2, ∞0) n X.) (f) Let X denote the set of real nụmbers with the one additional point a = (0, 1), and let the topology T for X consist of Ø and all subsets of X which contain a. Then there do not exist two disjoint non-empty open sets, so X is connected. Note that as a subspace of (X, T ), R is assigned the discrete topology and is therefore disconnected. This example (as well as examples (c), (d), and (e) above) demonstrate that the property of being connected is definitely not hereditary. %3D
Transcribed Image Text:(e) Let X = [0, 1] U [2, 3] with the subspace topology of the real line. Then A = [0, 1] and B = [2, 3] are disjoint, non-empty open subsets of X for which X = A U B, so X is disconnected. (Note that A and B are relatively open since A = (-∞, 3/2) n X and B = (3/2, ∞0) n X.) (f) Let X denote the set of real nụmbers with the one additional point a = (0, 1), and let the topology T for X consist of Ø and all subsets of X which contain a. Then there do not exist two disjoint non-empty open sets, so X is connected. Note that as a subspace of (X, T ), R is assigned the discrete topology and is therefore disconnected. This example (as well as examples (c), (d), and (e) above) demonstrate that the property of being connected is definitely not hereditary. %3D
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 9 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Propositional Calculus
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS