3. In each of the following examples, determine (with careful arguments) the bound- ary of A in X, and state whether A is open in X, closed in X, both open and closed in X, or neither open nor closed in X. b *X is any set with the discrete metric, A is any subset of X.

Q: 3.2 Prove that in any metric space (S, d) every closed ball S,xo is a closed set.

A: Let (S, d) be a metric space. Consider a closed ball Srx0 with both center x0∈S and radius r>0…

Q: 3. Let (X, d) be a metric space and x1, x2, ... , Xn, ... a sequenc

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Q: Theorem 3.7: A subset A ofa metric space (X, d) is closed if and only if /.contains all its limit…

A: We have to prove that A is Closed if and only if A contains all its limit points. Note : In proof i…

Q: Prove that in a metric space (S,d) every closed ball Sr[Xo] is a closed set

A:

Q: (1) Determine if each of the following statements is true(T) or false(F). 1- In a metric space,…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

Q: Let Y be an infinite subset of a compact set X c R. Prove Y' # Ø.

A: Y' denotes the derived set of Y i.e. the set of all limit points of Y. We just need to show that any…

Q: [3.12] Let S be a nonempty convex set in R", and let f. R" → R be defined as follows:

A: Let S be a nonempty convex set in Rn, and  Let f:Rn→R be defined as follows:…

Q: The only subsets of R which are both open and closed under the discrete metric are Ø and R. Select…

A: This is False statement, in discreate metric space, every subset of R is both open as well as…

Q: A metric space (X , d) and a collection of open sets U such that UEU is not open.

A: Let X, d be a metric space. Let U be the collection of open sets in the metric space X, d. That is,…

Q: Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

A:

Q: 3- In a complete metric space (X, d), If A C X such that, A = A then A is complete as a sub space of…

A:

Q: Every closed and bounded subset of X is compa The subset {z EX: |||| ≤ 1} of X is compact... X is…

A:

Q: (1) Determine if each of the following statements is true(T) or false(F). 1- In a metric space,…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

Q: Provethat amappi'ng from a metric SpaceXto metri'c space X is con tinuok ifand only f the inverse…

A:

Q: Prove if {Galpha}alpha e A is a collection of closed sets in a metric space, then…

A: Recall that arbitrary union of open sets in X is open in X.

Q: 3.2 Prove that in any metric space (S.) every closed ball Se[xo] is a closed set.

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Q: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

Q: 1.3 Classify each of the following sets in (R, d) ,where d is the usual metric on R, as open,…

A: Every open ball is an interval in R this leads prove for all parts.

Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

A:

Q: 6 Prove that a closed set in the metric space (S, d) cither is nowhere dense in S or clse contains…

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Q: 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

A:

Q: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

Q: 3.6 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

A:

Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

Q: Show that in R with the euclidean metric, any subset of R \Q with more than one element is not…

A:

Q: 4. Let (X, d) be a metric space and xnx where each x, e X and x e X. Let A be the subset of X which…

A:

Q: State whether each of the following statements is true or false, in either case…

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Q: Is the following statement True or False? Justify each answer. If S is a closed bounded subset of a…

A:

Q: (a) Suppose M is a set and d, d' are two different metrics on M. Prove that d and d' generate the…

A: Please find answer below:

Q: Let (N, d) be the discrete metric space, then the set A = {2,3, 4} is closed but not open open but…

A:

Q: 1.5 Let f : X → Y be a homeomorphism between metric spaces, the set U S X is closed in X if and only…

A: Since f:X→Y is a homeomorphism between metric spaces, it follows that f is continuous and bijective…

Q: Let E be a non-empty bounded subset of R. Prove that inf E is unique

A: Given: E is a non-empty bounded subset of ℝ. To show: inf E is unique.

Q: 3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a)…

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Q: Prove that If G Her 3-9 IS is an open set dense in the denx. S nowhere me tric space (sid)

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Q: The set S=xER:x²-4<0} with the usual metric is . O A. Compact. B. connected. C. Not connected D.…

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Q: State whether each of the following statements is true or false, in either case substan- tiate/…

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Q: 7 The set M={xEN:x<÷} with the usual metric is connected.

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Q: Let {An : n E N} be the indexed collection of sets defined by 1 2, 2+ 2n An Prove that: nMeN An = Ø

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Q: show by an example that a nonempty proper subset of S in the metric space (S, d) can be both open…

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Q: 3.6 Let A be a subset of a metric space (S, d). Show that if H is a nonempty open subset of S and An…

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Q: 2- In a metric space (X, d), A is the intersection of all closed subsets of X which contains A.

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Q: d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. Conclude that, with this metric,…

A: Given ℝ3,d is a metric space. To show that a subset S of ℝ3 is sequentially compact if and only if…

Q: a) Suppose that E is a set. If there exists a function f from E onto N, then E is at most countable.

A: (a) False f:E → INLet E= IR  Thenf: IR → INdefened as f(x) =  1  , ∨ x ∈ IRi.e There exist a…

Q: Consider X = {a1, a2, ..., an} Can a topology of X with 3 open and one with 4 open be equivalent?

A: We have to verify the given statement.

Q: 1- In a metric space (X, d), if A and A2 are open subsets of X then A1 U A2 is open in X. 2- In a…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

Q: 5. State whether the following is true or false m) A subset I of a metric space R with the usual…

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Q: )Š is the smallest closed set in M such that S CS.

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Q: 1. Let X be any nonempty set and let p E X, Prove that : E, {G X:p¢ G}U{X} is a topology on X.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: (b) (i) Give an example of an infinite collection of nested open sets in R under the usual metric on…

A: Here given usual metric on the set of real numbers R

Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

A: