4, 5 Let X be a metric space with metric d. Let r > 0 and define the open ball center atIEX with radius r>0 byB,(r) (ye X: d(r, u) 0), such that B(ro) C B}.1.Show that r is a topology in X.2.Show that B,(x) is open set with respeet to 7 for all r>0.3.Let r>0 and define the closed ball byB,(z) = {y € X : d(x, 4) Sr}Show that B,(r) is closed set with respect to T.4.Show that every open set with respect to 7 is a union of open ball(s).A point p is a limit point of E if for all r> 0, there is q # p such thatqE EnB,(p). For ECX, define E' be the set of all limit point of E. Show that E is5.closed if and only if E = EUE'.

Given X be a matric space with metric d. For any r>0, the open ball at center at x∈X with radius…

Answered: Show that every open set with respect…

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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Let (X,d) be a metric space and d*(x,y) = d(x,y)/1+d(x,y) Prove that the family of open sets with respect to the metric d is same as the family of open sets with respect to the metric d*.
Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B) ≠ ∅, prove that A∩∂ (B) ≠ ∅. Where ∂ (B) is the boundary of B.
Consider a set A and a function d: A × A → R that satisfies:• d(x, y) = 0 ⇔ x = y;• d(x, y) = d(y, x);• d(x, y) ≤ d(x, z) + d(z, y).Prove that (A, d) is a metric space, i.e. show that d(x, y) ≥ 0 forall x, y ∈ A.
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?
2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined byρ(x, y) = d(x, y)1 + d(x, y)for all points x, y ∈ S. Prove that (S, ρ) is a metric space, that it is bounded and thatρ(x, y) ≤ d(x, y) for all x, y ∈ S.
Let (X,d) be a metric space and E ⊆ X. Prove that if E is compact, then E is bounded.
Let (X,d) be a metric space. For x,y in X define e(x,y)=min{1,d(x,y)}. Prove that (X,e) is also a metric space.
Is the set S = [0,1] with the discrete metric d separable? Explain.
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).]
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?
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(using Euclidean metric). Is H bounded?
the usual metric space defined by d(x,y)= x-y prove the four propertis of metric space

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Show that every open set with respect to 7 is a union of open ball(s). A point p is a limit point of E if for all r 0, there is q p such that qE En B,(p). For ECX, define E' be the set of all limit point of E. Show that E is 4. 5. closed if and only if E EUE'.