4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains some nonempty open set.

Q: Show in the discrete metric space: 1. All sets are open (especially single points);

A: Let X be a non-empty set and d be discrete metric on X

Q: 5. If G1,G2 be two given closed sets in a metric space (X, d) prove that: G1 U G2 is closed.

A: We have to prove given result:

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

A:

Q: 2.4.5. Let X be a metric space. Given x E X, prove that {x} is a closed subset of X.

A: Let (X,d) be a metric space to show that {x} is closed sunset of X.

Q: Show with all the relevant details, albeit succinctly that every compact metric space M is complete.

A: Here we use sequential definition of compactness so that we use that in proving the completeness of…

Q: 3.1 Show that an infinite intersection of closed sets E k = 1,2,3,., in a metric space (SA) is a…

A:

Q: Let (X, d) be a metric space with X being an infinite countable set. Show that X is not connected.

A: A set is said to be infinite countable if its elements can be counted like natural number but you…

Q: let A be a bounded. Sup A non. n-empty set in R show that is a limit point of A.

A:

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: 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: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

A: T2-space: A topological space X is said to be T2-space for given x, y∈X, there exists open sets U…

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

A:

Q: Prove or disprove. A closed and bounded subset of e' is compact.

A: PROVE OR DISPROVE :-  A closed and bounded subset of l1 is compact.

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

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

A:

Q: A normed space X overR is separable if it has a conutable dense subset 0 صواب ihi

A: The given statement that " A normed space X over R is seperable if it has a countable dense subset "…

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

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Q: Show in the discrete metric space: 1. All sets are open (especially single points); 2. All…

A: We first define Discrete space

Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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Q: 3.5 Show that the metric space (R, d), where d(r, y) = |x – y|, is separable.

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

Q: 3. Prove that if G is an open set dense in the metric space (S, d) then SG is nowhere dense in S.

A:

Q: Prove the following analogue of Theorem 3.10(b): If {Ein} is a sequence of closed nonempty and…

A: given,En is a sequence of closed nonempty and bounded sets in a complete metric space X. En⊃En+1 and…

Q: 3.1 Show that an infinite intersection of closed sets F, k = 1,2, 3, ..., in a metric (S, d) is a…

A: Let {Fk}  be an arbitrary collection of closed sets . . 1  i.e. for k = 1, 2, 3, . . . . . . . . To…

Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

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

A:

Q: 2.5 For each of the following properties, give an example consisting of two subsets X, Y CR', both…

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Q: 2.6.13 LetE be a subset of a metric space X. Prove that E has empty interior (E° = Ø) if and only if…

A: Dense set and interior set

Q: 4. Let {Ka}aeA be a collection of compact subsets of a metric space, Show NaEA K, is closed

A:

Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

Q: 3. Let K,and K, be compact subsets of a metric space X. Show K, U K2 is compact

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Q: ã) In general, for any metric space (M, d), we have 1) Sis compact in M = S is closed and bounded in…

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

Q: Let S be a non-empty open subset of R that is bounded above. Show that sup S exists but that max S…

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Q: If A and B are bounded subsets of R, then AuRand AOR are bounded: prove that

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Q: Show that any compact set in R is closed and bounded.

A: Proof

Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

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Q: (a) Prove that every closed subset S of a compact metric space (M,d) is compact in M.

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

Q: Prove the following: Let (Y, d) be a metric space. If Y is totally bounded, then Y is bounded.

A: Given:  (Y,d) is a metric space, and Y is totally bounded To Prove Y is bounded Definition: Let…

Q: Show that the closed ball Y in a metric space (x,d) is a closed set .Also show that if (x,d) is…

A:

Q: Recall that RK denotes the real line with the K-topology. Let Y denote the quotient space obtained…

A: Given:-             Note that if U opens in R, and then U-K is opens in R k.             U = union…

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: Prove that in any metric space (S, d) every closed ball S,[ro] is a closed set.

A: Use the following definitions, to prove the result. If a set contains all its interior points, then…

Q: Let M, be set of all rare subsets of a metric space X. Prove that countable union of M, is rare.

A: As per bartleby guidelines for more than one question asked only first should be answered. Please…

Q: Show that all open balls are bounded subsets of a metric space.

A: Solution: Let M=X,d be a metric space. For an arbitrary point a∈X, the open ball Bεa where ε>0 is…

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: A metric space is said to be separable if it contains a countable dense subset. (a) Is R separable?…

A: Set of all rational numbers Q belongs to R and Q is dense in R . Also set of all rational numbers Q…

Q: 2) Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

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Q: 1/ For any. topological space (X, T), if T has only three elements then (X, T) is connected.

A: Given: Topological space X, T and T has 3 elements  To prove that X, T is connected when T has 3…