Exercise 1. [Metric] Let p be a prime number, and d.: ZxZ→ [0, +∞[ be a function defined by -max(meNp divides (z-9)) dp(x, y) = p ma Prove that d, is a metric on Z and that d,(x, y) < max{d₂(x, 2), dp(z,y)}, for every x, y, ZE Z
Exercise 1. [Metric] Let p be a prime number, and d.: ZxZ→ [0, +∞[ be a function defined by -max(meNp divides (z-9)) dp(x, y) = p ma Prove that d, is a metric on Z and that d,(x, y) < max{d₂(x, 2), dp(z,y)}, for every x, y, ZE Z
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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topology exercice 1
![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 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%2F17e6903a-a4b4-4496-b3ca-36a828fdb1e6%2Fwvuto0e_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(mɛNp™ 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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