Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Exercise 6. [Topology] Let X be a nonempty set and 7 = {U C; U = 0 or X\U is countable}. Prove that is a topology on X.

Exercise 6. [Topology] Let X be a nonempty set and 7 = {U C; U = 0 or X\U is countable}. Prove that is a topology on X.

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
Question
topology question6
Exercise 1. [Metric] Let p be a prime number, and d: ZxZ→ [0, +∞[ be a function defined by dp(x, y) = p-max(mɛNp™ divides (2-3)) Prove that d, is a metric on Z and that dp(x, y) < max{dp(x, 2), dp(z,y)}, for every x, y. ZE Z Exercise 2. [Closed in metric space] Let (X, d) be a metric space and F € X be a finite subset. Prove that F is closed in X. Exercise 3. [Closure in metric space] Let (X, d) be a metric space and Y be a nonempty subset of X. The distance of a point X from the subset Y is a function X→ [0, +∞[ defined by d(x, y) = inf{d(x, y); y = Y}. 1. Verify that the distance function is well defined. 2. Prove that Y = {x € X; d(x, y) = 0}. Exercise 4. [Separable space] Let X be a set of all real sequences (zn)neN converging to 0. Prove that the function d: X X X → [0, +∞o[ (In, Yn) → d(In, Yn) = sup |In - yn NEN is a metric on X. Show that the metric space (X, d) is separable. Exercise 5. [Restriction of metric] Let (X,d) be a metric space and U be a proper open subset of X. Consider a function du : U x U→ [0, +∞[ 1 d(x, X\U Prove that du is a metric on U and that it is equivalent to the induced metric duxu. (x,y) → dv(x, y) = d(x, y) + 1 d(y, X\U Exercise 6. [Topology] Let X be a nonempty set and 7 = {U C; U = 0 or X\U is countable}. Prove that is a topology on X.
Transcribed Image Text:Exercise 1. [Metric] Let p be a prime number, and d: ZxZ→ [0, +∞[ be a function defined by dp(x, y) = p-max(mɛNp™ divides (2-3)) Prove that d, is a metric on Z and that dp(x, y) < max{dp(x, 2), dp(z,y)}, for every x, y. ZE Z Exercise 2. [Closed in metric space] Let (X, d) be a metric space and F € X be a finite subset. Prove that F is closed in X. Exercise 3. [Closure in metric space] Let (X, d) be a metric space and Y be a nonempty subset of X. The distance of a point X from the subset Y is a function X→ [0, +∞[ defined by d(x, y) = inf{d(x, y); y = Y}. 1. Verify that the distance function is well defined. 2. Prove that Y = {x € X; d(x, y) = 0}. Exercise 4. [Separable space] Let X be a set of all real sequences (zn)neN converging to 0. Prove that the function d: X X X → [0, +∞o[ (In, Yn) → d(In, Yn) = sup |In - yn NEN is a metric on X. Show that the metric space (X, d) is separable. Exercise 5. [Restriction of metric] Let (X,d) be a metric space and U be a proper open subset of X. Consider a function du : U x U→ [0, +∞[ 1 d(x, X\U Prove that du is a metric on U and that it is equivalent to the induced metric duxu. (x,y) → dv(x, y) = d(x, y) + 1 d(y, X\U Exercise 6. [Topology] Let X be a nonempty set and 7 = {U C; U = 0 or X\U is countable}. Prove that is a topology on X.
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
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,
SEE MORE TEXTBOOKS