Exercise 4.6. (1) Consider R² with the standard topology. Let p ER² be a point not in a closed set A. Show that inf{d(a, p) | a E A} > 0. (Recall that inf E is the greatest lower bound of a set of real numbers E.) (2) Show that R² with the standard topology is regular. (3) Find two disjoint closed subsets A and B of R² with the standard topology such that inf{d(a, b) | a E A and b e B} = 0. (4) Show that R² with the standard topology is normal.
Exercise 4.6. (1) Consider R² with the standard topology. Let p ER² be a point not in a closed set A. Show that inf{d(a, p) | a E A} > 0. (Recall that inf E is the greatest lower bound of a set of real numbers E.) (2) Show that R² with the standard topology is regular. (3) Find two disjoint closed subsets A and B of R² with the standard topology such that inf{d(a, b) | a E A and b e B} = 0. (4) Show that R² with the standard topology is normal.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Could you explain how to show 4.6 in easiest possible way(in very detail)?
Transcribed Image Text:Exercise 4.6. (1) Consider R² with the standard topology. Let pER² be a point not in
a closed set A. Show that inf{d(a, p) | a E A} > 0. (Recall that inf E is the greatest
lower bound of a set of real numbers E.)
(2) Show that R² with the standard topology is regular.
(3) Find two disjoint closed subsets A and B of R² with the standard topology such that
inf{d(a, b) | a E A and b E B} = 0.
%3D
(4) Show that R² with the standard topology is normal.
Definition. Let (X,J) be a topological space.
(1) X is a T1-space if and only if for every pair x, y of distinct points there are open sets
U,V such that U contains x but not y, and V contains y but not x.
(2) X is Hausdorff, or a T2-space, if and only if for every pair x, y of distinct points
there are disjoint open sets U,V such that x E U and y e V.
(3) X is regular if and only if for every point x E X and closed set A C X not containing
x, there are disjoint open sets U,V such that x € U and A C V. A T3-space is any
space that is both T and regular.
(4) X is normal if and only if for every pair of disjoint closed sets A, B in X, there are
disjoint open sets U,V such that A C U and BC V. A T4-space is any space that
is both T and normal.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 6 steps
Knowledge Booster
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.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
