topology exercice 1 Exercise 1. [Metric] Let p be a prime number, and d: ZxZ→ [0, +∞[ be a functiondefined bydp(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 ZExercise 2. [Closed in metric space] Let (X, d) be a metric space and F € X be afinite 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 nonemptysubset of X. The distance of a point X from the subset Y is a function X→ [0, +∞[defined byd(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 convergingto 0. Prove that the functiond: X X X → [0, +∞o[(In, Yn) → d(In, Yn) = sup |In - ynNENis 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 properopen subset of X.Consider a functiondu : U x U→ [0, +∞[1d(x, X\UProve 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) +1d(y, X\UExercise 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.

Answered: Image /qna-images/answer/17e6903a-a4b4-4496-b3ca-36a828fdb1e6.jpg

Answered: Exercise 1. [Metric] Let p be a prime…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 27CR: Verify the triangle inequality and the Cauchy-Schwarz Inequality for uandv from Exercise 25. Use the...

See similar textbooks

Similar questions

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.
Verify the triangle inequality and the Cauchy-Schwarz Inequality for uandv from Exercise 25. Use the inner product given in Exercise 25. 25. For u=(4,32,1) and v=(12,3,1), a find the inner product represented by u,v=u1v1+2u2v2+3u3v3 and b use this inner product to find the distance between uandv.
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.
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.
1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {ϕt}) be a flow on a metric space X. When is xo in x a fixed point of the flow? c) When do you say that a fixed point xo in x is Poincare stable? d) When do you say that a fixed point xo is Lypanov stable?
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.

SEE MORE QUESTIONS

Recommended textbooks for you

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS

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