Give an example that satisfies in arbitrary metric space, boundedness and closedness do not necessarily imply compactness.

Q: Prove that any contraction between metric spaces has at most one fixed point

A: According to the given information, it is required to prove that any contraction between metric…

Q: Q7: Prove that (R", |l-|| ) is complete metric space.

A: This is a problem of Topology, Functional Analysis.

Q: Remark. For everyr E N, a set in R" with the Euclidean metric is com- pact if it is bounded and…

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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: 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: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

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Q: show that any infinite discrete topological space is not compact

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Q: Every totally  bounded metric space is compact

A: This statement is not always true.

Q: Let (R. d) be discrete metric space then R is not complete O True O Faise

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Q: Prove that linear maps are bounded for multivariable calculus

A: In this question, concept of linear map is applied. Linear Map The matrix of the linear map can…

Q: Please define these cocepts: Isolated point, .discrete topology, dense space

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Q: A completely regular space is also T1-space 1 True 2 False

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Q: 5. Prove that the property of a space having an isolated point is a topological property. 6. For any…

A: Sol

Q: Let (R,d) be diserete metric space then R is not compact . True or false.??

A: Let (R,d) be diserete metric space then R is not compact . True or false.??

Q: Show that recel projeetrue space is an manifol a.

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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 metric space is sequentially compact if and only if it is totally bounded and complete. prove it

A: We need to prove that any given metric space is sequentially compact if and only if it is totally…

Q: What is norm space

A: In mathematics, a normed space or a normed vector space is a vector space over the real or complex…

Q: Find an example of a topological space that is seperable and first countable but not second…

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Q: Exercise. Every point in a discrete metric space forms an open ball.

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Q: Show that: All completions of a metric space are isometric.

A: According to the given information, it is required to show that all the completions of a metric…

Q: Show that ℓ^1 is a normed linear space.

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Q: Every inner product space is metric space and normed space O metric space but may be not normed O…

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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: Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both…

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Q: Exercise. A finite set in a metric space cannot have accumulation points.

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Q: Show that the compactness in satability theorem is neccessary.

A: The Stability Theorem fails when X is not compact

Q: Is it possible to have a function on a metric space which is discontimuous at every point of the…

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Q: A finite set in a metric space cannot have accumulation points.

A: We have to prove that A finite set in a metric space can not have accumulation (Limit) point.

Q: Q7: Prove that ( IR", ||·|| ) is complete metric space.

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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: مممممم م العمل لها -Pre-Hilbert space - Metric space and Complete - Normed space and complet

A: 1) the banach space is solution definition of Banach space A Banach space is a normed linear space…

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

A: This is a problem of Functional Analysis.

Q: (a) Prove that every nonempty proper subset of a connected space has a nonempty boundary. Is the…

A: As per Bartleby guidelines, for more than one question asked, only first is to be answered. Please…

Q: What is the Boundedness Theorem?

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Q: Show that topologically completeness is weak hereditary property.

A: In this question, we have to show that completeness is a weak hereditary property.

Q: 3: Prove that any reflexive normed space is complete.

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Q: Prove that a finite dimensional normed linear space is always reflexive.

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

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Q: Prove that a normed linear space is complete if its unit sphere is complete.

A: We prove that a normed linear space is complete if its unit sphere is complete. Its straightforward…

Q: Q7: Prove that ( IR", II-||) is complete metric space.

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Q: a) Find a space that is a regular but not Hausdorff. Justify your claim.

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

Q: Every compact, Hausdorff space is normal.

A: To prove: Every compact, Hausdorff space is normal. Let X be a compact Hausdorff space. Let A,B⊂X be…

Q: 2.2 Is any discrete space complete? Give an example to substantiate your answer.

A: Yes, any discrete space is complete.

Q: Let (X, τ ) be a discrete space. Is (X, τ ) second countable? Explain why

A: Consider the provided question, Let x,τ be a discrete space,    ⇒x,τ  be denote topology if x,τ is…

Q: Is any discrete space complete? Give an example to substantiate your answer.

A: A metric space is complete if and only if every Cauchy sequence is convergent.

Q: the euclidean space R³ is a separable metric space. true or false?

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…

Q: Show that the closure of a connected space is connected.

A: