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 : Ux U→ [0, +∞[ 1 d(x, X\Ud(y, 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) +
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 : Ux U→ [0, +∞[ 1 d(x, X\Ud(y, 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) +
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
topology exercice 5
![Exercise 1. [Metric] Let p be a prime number, and d: ZxZ→ [0, +∞[ be a function
defined by
dp(x, y) = p-max(mEN;" divides (2-3))
Prove that d, is a metric on Z and that d,(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 (n)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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbc44ee4c-f0ee-41b9-a7bf-13469d6ba446%2F6ba6a3c0-46d2-4c82-b1ba-acf7d2ac1e7b%2F9clhpsg_processed.jpeg&w=3840&q=75)
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(mEN;" divides (2-3))
Prove that d, is a metric on Z and that d,(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 (n)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.
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