Show that a union of any finite points of bounded subsets of a metric space is bounded

Q: Define the closure of a sot I in a motric spack. How is the boundry it a sut I related to it clogwai

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Q: a) Show that every closed subset of a compact space is compact but the converse need not to be true…

A: As per the guidelines I had given the proof of the first theorem. Kindly repost the another theorem…

Q: Exercise 3.55. Show that every wedge sum of Hausdorff spaces is Hausdorff.

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Q: Prove that any Space is complete. finite dimensional normed

A: Introduction: Completeness is a property of a topological space. When in a topological space, every…

Q: Let X be a locally compact Hausdorf f space

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Q: UNA annetect in X show that Tivi is a connected set in a space Y. ) Let T be a cofinite topology…

A: If  X is a non-empty set, then the collection of subsets of X whose complements are finite along…

Q: Suppose that X and Y are topological spaces and f: X- Y. Show that f is continuous iff for each…

A: Definition: Let X and Y are topological spaces and f : X→Y be a function. We say that the function f…

Q: Q1\ Define the compact set. Then, prove that: if A is a compact subset of F , there exists a best…

A: Compact set : A set is compact if and only if every open cover has a finite subcover. There is also…

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

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Q: * Every Pre-Hilbert space is I - Continues Normed space -orthogonal Metric space inner Produced

A: Every pre-Hilbert Space is a complex vector space X together with an inner product (A complex inner…

Q: Ex: Let d is a trivial metric o space ) show that every infir

A: We have given trivial metric space d on infinite set X. We have to show that every infinite subset…

Q: 2. Define the concepts of (a) first countable space; (b) second countable space; (c) separable space…

A: According to Bartleby guidelines I have explained defination of first countable space, second…

Q: Show that finite dimensional inner product space is Hilbert space

A: We are to show that finite dimensional inner product space is Hilbert space.

Q: Don't use chat bot     Show that a compact Hausdorff is metrizable if it is locally metrizable.

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Q: Exercise 3. In this exercise (X, d) is a metric space. .} 1) Assume that X is compact. For every r>…

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Q: cise 2 Let X be a topological space. - Show that X itself is a Baire space. T

A: Given: X is a topological space. To Prove: We have to show that the topological space X itself is a…

Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology.   We claim that Z is NOT…

Q: let ( X Td)be Toplog9 me trics spaces the n EVery singteon set is closed set

A: Given that X,Td is a Topology Metric spaces.

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: With the help of exomples Show thaat on finite dimensional are equivalent. any two norms Linear…

A: Given: Any two norms on finite dimensional linear space are equivalent.

Q: Theorem 3.5. Finite Intersection Property If {Ka} is a collection of compact subsets of a metric…

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Q: Suppose X is a connected topological space with the property that every point x of X has a…

A: In this problem, we consider a connected topological space X with the property that every point in X…

Q: 1. A non-empty complete and bounded metric space (X, d) with exactly a countably infinite (not…

A: To Find: An example of a complete and bounded metric space (X,d) with countably infinite number of…

Q: Hausdorff space.

A: Given: A set X with co-finite topology and  Hausdorff space. To Find: Under what condition a set X…

Q: (iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the…

A: Given : X be a set with atleast two points. To prove : Discrete space and indiscrete space ie. (X,τ)…

Q: ## Commpact subse regular Space X and if of is subset of A such that .cey th there is a closed a…

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Q: Exercise 1. Let X be a metric space. The subsets of X which are arbitrary (finite or infinite)…

A: What is Open Set: Open sets in a metric space are those that, for every point P, include all other…

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

A: Topology Question

Q: 1. Show that any interval (a,b) in R with the discrete metric is locaaly compact but not compact

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

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

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Q: determine an b) Every Space CD Every non True or False, Justify why, not totally bounded Open Set S…

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Q: Prove that ℝ with lower limit topology is Hausdorff

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Q: Let (M, d) be a metric space and K ⊂ M. The following statements are equivalent: (i) K is relatively…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Let (C[a,b] .d) be the discrete metric space. Then Cla,b] is bounded. * False True

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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: True False An open interval is not homeomorphic to either a closed interval or a half open interval.…

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Q: 9 Ex: Show that every finite space is compact space. Eulet die a trivial matrie on infinite catV(Y…

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Q: ( 1) Give an example of a compact space which is not Hausdorff. 2) Give an example of a locally…

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Q: SHOW that the continuous image Path - connected space is path Connected.

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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: quutiont of a compact space is compact and connected Show that connected

A: Show That :  The quotient of a compact (connected) space is compact (connected).