Detailed help please Let (X, d) be a metric space and E a non-empty subset of X. An element x = Xis said to be on the boundary of E if every neighbourhood Nr(x) of x containsa point in E and a point not in E. The set consisting of the boundary pointsof E is denoted by E. Prove(a) E is closed if and only if JE ○ E.(b) E is open if and only if EnæE = Ø.(c) E = EUOE.

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Answered: Let (X, d) be a metric space and E a…

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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(c) Prove that a metric space is connected iff it contains exactly two sets that are both open and closed.
Prove that if X,d is a metric space and G ⊆ H ⊆ X, then G-bar ⊆ H-bar.
Show that the interval [0,1] is a closed set in R.
5. If G1,G2 be two given closed sets in a metric space (X, d) prove that: G1 U G2 is closed.
1. Prove: Let X and Y be two topological spaces. Then the function f:X →Y is continuous if and only if inverse image of every closed set is closed in X. Give an example to substantiate the proof.
Define metric space and metric topology on X. Prove that in a metric space (X, d) (i) & and X are open sets. (ii) the union of arbitrary collection of open sets is open. (iii)the intersection of finite number of open sets is open. Justify by an example that countable infinite Intersection of open sets may notbe open.
Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only if every neighborhood of x contains a point of A.
5. State whether each of the following statements is true or false, in either case substan- tiate/ justify. such that (c) The set X = [3, 9] as a subset of R with the discrete metric is disconnected.
3.2 please
Ex: Let d is a trivial metric on infinite set X ((X,d) is infinite trivial metric space ) show that every infinite subset of X is not compact set.
4. Let x be an interior point of a subset S of a metric space (X,d). Show that x must also be a limit point of S.
Theorem 4.4.8. Let (S, d) be a metric space. A set A is open if and only if its complement R\A is closed. We can interpret this Theorem in the following way: • complements of open sets are closed • complements of closed sets are open For a

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