1) In a metric space (M, d), show that both 0 and M are open sets in M. 2) In a metric space (M, d), show that every open ball BM(a; r) is an open set in M.

Q: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

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Q: Is R, equipped with the finite-closed topology, Hausdorff? Provide a proof supporting your answer.

A: Given: ℝ is equipped with the finite-closed topology. Let, ℝ with this topology be Hausdorff and 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…

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A: Consider the provided question, We have to prove by using only definition of a compact set, that (0,…

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

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

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

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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…

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Q: In a metric space (M, d), show that both Ø and M are open sets in M. In a metric space (M, d), show…

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Q: Let (X, I) be a topological space and suppose that A and B are subsets of X such that ACB. Prove…

A: Let x ∈A°. Then by definition, there exists an open set U  such that  x∈U⊆A.Since A⊆B hence x∈U⊆B ,…

Q: 1) In a metric space (M, d), show that both Ø and M are open sets in M. 2) In a metric space (M, d),…

A: Open sets :- Let (M, d) be a metric space then a subset A of M is open if and only if each point of…

Q: A subset I of a metric space R with the usual metric is compact if and if only it is an interval…

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

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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: Let d and e be metrics on a set X such that for each ball Ba center at pE X there exists a ball Be…

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Q: Be M, N subsets of space W. Then span (MUN) = span (M) +span (N)- %3D

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Q: Let

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Q: Let (X,d) be any metric space, then the only open sets and closed sets are and X O True O False

A: Given statement is false because if we take descrete metric space

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If ACB,…

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Q: 1) In a metric space (M, d), show that both 0 and M are open sets in M. 2) In a metric space (M, d),…

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Q: Let (Ka)aeA be a collection of compact subsets of a metric space, X. Show naeAKa is closed in X.

A: A metric space X is compact if every open cover of X has a finite subcover.

Q: Let (X, T ) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f ∈ C(X, M);…

A: Given as X,T be a topological space,M,d be a complete metric space and BCX,M=f∈CX,M;fX is bounded…

Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

Q: Given a metric space M with metric d, verify that any ε-ball is an open set.

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Q: The set S=[1,3) is not compact. Which open cover of S contains a finite subcover?

A: This is the question of Topology.

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: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. biso (2) If…

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

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

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Q: Let (R, d) be discrete metric space, then R is not separ able O True O False

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

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

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Q: Prove that in any metric space, an ɛ- eighborhood of a point is an open set.

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Q: Amongst the various possible products of four topological spaces A, B, C and D, show that (A × (B ×…

A: We shall solve this in next step

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: Task 13 Let (M, d) be a compact metric space. Show that closed subsets of M are compact.

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Q: Prove that in a metric space (X, d) every closed ball that is a set K(x, r) = {y e X : d(x, y) <=…

A: Question: Prove that in a metric space X, d every closed ball K¯x, r=y∈X, dx, y≤r is a closed set.…

Q: Show that the set of all real numbers constitutes an incomplete metric space if we choose d(x, y) =…

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

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: Given a metric space M with metric d, verify that any E-ball is an open set.

A: Given that M is a metric space with metric d. Let Ba,ε be an ε-ball with center at a. That is:…

Q: Exercises: 1) In a metric space (M,d). show that both Ø and M are open sets in M.

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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: (b) Prove that the intersection of finite collection of open sets in the metric space (M, d) is open…

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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…

Q: If A is a compact subset of a metric space (X, d) and B is a closed subset of A, prove that B is…

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Q: Let E and F be compact sets in a metric space (X,d). Show that E U F is compact using the definition…

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Q: 3.3 Let 11 and ra be distinct points in the metric space (S, d). Verify that there are open balls…

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