Answered: Exercise 3. [Closure in metric space]… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.3: Vectors

Problem 60E

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Question

topolgy exercice 3

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, z), 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 ze 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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Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Prove that topological space E is not homeomorphic to the spaceY = {(x, y) ∈ E^2 : y = ± x} (E represents R equipped with Euclidean distance, E^2 represents R^2 equipped with euclidean distance)
A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?
A. Let H be the set of all points (x, y) in ℝ2 such that x2 + 3y2 = 12. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?
(a) Supply a definition for bounded subsets of a metric space (X, d). (b) Show that if K is a compact subset of the metric space (X, d), then K is closed and bounded. (c) Show that Y ⊆ C[0, 1] from Exercise 8.2.9 (a) is closed and bounded but not compact.
Can you prove in detail the metric space symmetry condition of the discrete metric, I find myself writing that if d(x,y)=0 and x=y then y=x which means that d(y,x)=0. But I have to prove it in a detailed manner with explanation apparently. Thank you in advance.
the usual metric space defined by d(x,y)= x-y prove the four propertis of metric space
1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {phi_t}) be a flow on a metric space x. what is x0 in x a fixed point of the flow? c) When do you say that a fixed point x0 in x is Poincare stable? d) When do you say that a fixed point x0 is Lyapunov stable? Use Analysis to complete the following statements.
Consider ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. i. Verify that d is really a metric.ii. State and prove the Bolzano-Weierstrass Theorem for ℝ3 with this metric.
Let (X,d) be a metric space with the added condition that for any three points x,y,z in X, d(x,y) <= max{d(x,z),d(y,z)}.(a) Show that every triangle in X is isoceles.(b) An open ball in X with center u in X and radius r > 0 is defined as B(u;r) = {x in X | d(u,x) < r}. Show that every point in an open ball is a center for the open ball. [Hint: Part of your argument might include showing that if v in B(u;r), then B(v;r) = B(u;r).]
Let (R>0, d) be the metric space defined by d(x, y) =|log (y/x)|. This metric space is isometric to the Euclidean line E1, where an isometry E1 → (R>0, d) is given by x→ ex . proof that x→ ex is isometric.
A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?(Please Use As much text as you can so the explanation is a bit more detailed)

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