Let A be a sSubset of a metric space A. Given r€A, define the distar from x to A to be dist(z, A) = inf{d(z, y) : y € A}.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
icon
Concept explainers
Topic Video
Question
Material :Daly analysis
2.4.7. Let A be a subset of a metric space X. Given r E X, define the distance
from r to A to be
dist(r, A) = inf{d(r, y) : y E A}.
2.4 Closed Sets
61
Prove the following statements.
(a) If A is closed, then r E A if and only if dist(r, A) = 0.
(b) dist(r, A) < d(r, y) + dist(y, A) for all r, y E X.
(c) dist(r, A) – dist(y, A)| < d(x, y) for all r, y E X.
Additionally, show by example that it is possible to have dist(r, A) = 0 even
when z ¢ A.
II
Transcribed Image Text:2.4.7. Let A be a subset of a metric space X. Given r E X, define the distance from r to A to be dist(r, A) = inf{d(r, y) : y E A}. 2.4 Closed Sets 61 Prove the following statements. (a) If A is closed, then r E A if and only if dist(r, A) = 0. (b) dist(r, A) < d(r, y) + dist(y, A) for all r, y E X. (c) dist(r, A) – dist(y, A)| < d(x, y) for all r, y E X. Additionally, show by example that it is possible to have dist(r, A) = 0 even when z ¢ A. II
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Knowledge Booster
Sample space, Events, and Basic Rules of Probability
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
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,