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

Q: Show that every second countable (A2-space) space is separable.

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Q: Suppose A and B be subsets of metric space. Show that if A is compact and B is closed then A + B is…

A: Given A and B are subsets of a metric space. and also Given A is compact , and B is closed. We have…

Q: Let (X,T) be a toplogical space, A, BC X. Show that АС В Ас в, АсВ. 介

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Q: Let (N, d) be the usual metric space, then the set A= {1,3} is O closed only O neither open nor…

A: ( N,d) be a usual metric space of natural number.  A={1,3} be a finite subset of N. Since A is…

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: are closed.

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Q: Show that every countable subset of R" has Hausdorff dimension 0.

A: claim- to show that every countable subset of ℝn has Hausdorff dimension 0. it is known that…

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: Let E1, E2, ..., En ⊂ X,d be compact subsets of a metric space X. Just using the definition of…

A: For a metric space we know that the for a compact set every open cover has a finite subcover. So…

Q: Let (X, d) be a metric space and let A ⊆ X be complete. Show that A is closed.

A: Given that X,d be a metric space and A⊆X be complete. The objective is to show that A is closed.

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

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

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

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Q: (a) Prove that every finite subset of a metric space is closed and has umulation points.

A: Since you have posted multiple questions, but according to guidelines we will solve the first…

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: Any cover of a metric space is open.

A: Any cover of a metric space is open_____?_____

Q: Exercise. Prove that in a metric space (X, p), a set G C X is open if and only if it is a union of…

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Q: Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact

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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…

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

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Q: (b) Show that the metric space (Q,1) is not complete. (c) Prove that the discrete metric space (M,d)…

A: Complete Metric Spaces :- A metric space (X, d) is said to be complete iff every Cauchy sequence in…

Q: 7. If in an inner product space, (x, u) =(x, v) for all x, show that u = v.

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Q: in the discrete metric space: 1. All collections are clo

A: Let (X, d) be a metric space. Consider a closed ball Brx0 with both center x0 ∈ X and radius r >…

Q: Let (X, d) be a metric space. a) Assume that A is a (finite or infinite) collection of open sets.…

A: To orove the required topological properties of open sets in a metric space (X,d)

Q: If x is a limit point of a subset A of a metric space X, then sh

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Q: If A is a subset of a metric space X, then show that A = (x: each neighbourhood of x intersects A}

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Q: every finite subset of metric space is compact. ture or false?

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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 metric. Closed ball in space closed a (X,d) Also, show Complete is Complete. a. set.…

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

Q: Prove that (R", II-|) is complete metric space.

A: This is a problem of Functional Analysis.

Q: Exercise. Prove that in a metric space (X,p), a set G CX is open if and only if it is a union of…

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Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

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Q: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

Q: Prove that if Gn is an open set which is dense in the complete metric space (S,d) for n = 1,2,...…

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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: Every closed subset of a connected metric space is connected. O True O False

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Q: Let (R, d) be discrete metric space and A = N then A is complete and compact neither complete nor…

A: Given that (ℝ,d) is a discrete matric space and A=ℕ. then     d(x,y)=0 if x=y1  if x≠y A matric…

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: Prove that if X,d is a metric space and G ⊆ H ⊆ X, then G-bar ⊆ H-bar.

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Q: (a) Show that every metric space is T2-space.

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Q: Let X be ametric space. If A and B are each dense open subsets of X then AOB is dense open subset of…

A: First we have shown the intersection is open. Then we use the property that if a set intersect every…

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: Exercise. Prove that in a metric space (X, p), a set GC X is open if and only if it is a union of…

A: Solution

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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