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

Q: Let X = [0, 1] U [2, 3] endowed with the standard metric. Then [0, 1] is an open subset of X. Select…

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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: 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, 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: Prove that if the space X' of a normed linear space X is separable (as a metric space) then X itself…

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Q: Suppose that l is a line. Let H be one of the two half-planes bounded by l. Give a proof that l U H…

A: Answer is mentioned below

Q: (b) Prove that the set S = {x EQ: √2<x<√3} is closed and bounded in (Q.1-1), but not compact in…

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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: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded. %3D

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Q: Let H be the set of all points (x, y) in ℝ2 such that x2 + 3y2 = 12. Show that H is a closed subset…

A: Please check the answer in next step

Q: 3. Show that X = [0; 1] with the discrete metric is bounded but not totally bounded

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Q: Show that X = [0, 1] with the discrete metric is bounded but not totally bounded.

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Q: The set [0, 1] with the discrete metric is compact. True False

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Q: Consider IR as a metric space.with metric dux, y) = 1x-y1. Show that the open interval C0, 1) is an…

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

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Q: Exercise 5. 1) Show that the metrics dp, p ≥ 1 and dmax on Rª are all Lipschitz equivalent. 2) If X…

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Q: Find all limit points of a set A = {(2+,2 – |n e N} on the Euclidean plane with the isual metric. Is…

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Q: Prove that every finite T,- space has the discrete topology.

A: T=(X, τ)  (X, τ) is a T1 space if and only if all points of X are closed in T.   Let X≠∅ be a set.…

Q: Show that in R with the euclidean metric, any subset of R \Q with more than one element is not…

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Q: If there exists open ball B (a, r) such that B (a, r) nG=Ø then a is not limit point of G

A: To check whether there exists open ball B(a, r) such that B(a, r) ∩ G= ∅, then whether a is not…

Q: 2.5 Show that any interval (a,b) inR with the discrete metric is locally compact but not compact.

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Q: The set R- {0} with the usual metric is connected.

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Q: 1. Show that the real line is a metric space.

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

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Q: Show that every closed ball in R" is an n-dimensional manifold with Assuming the theorem on the…

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

A: 1. Let E and F be compact subsets in a metric space X,d. Let UE and UF be arbitrary open covers of E…

Q: Prove that ℝ with lower limit topology is Hausdorff

A: Solution :-

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

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Q: Let K be a compact subset of a metric space X, and assume that -R is continuous. Prove that f…

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

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

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Q: 1) Prove that for any space, X, with any topology, T, that: (i) CI(X) = X and (ii) Int(X) = X.

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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: Let x be an interior point of a subset S of a metric space (X,d). Show that x must also be a limit…

A: Interior Point: Let A be a subset of metric space X, d. A point x∈A is called an interior point of A…

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 by sequences that the open balls in R^n with the usual (Euclidean) metric are not compact

A: In this question, we have to show that open balls in ℝn with the usual metric are not compact by the…

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

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Q: For a metric space (X, d) and a continuous functionf from X into itself, show that the set of points…

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Q: Use the Heine-Borel theorem to show that every closed and bounded interval [a, b] with the absolute…

A: Compact Set: A collection A of subsets of space X is said to be cover X or covering of X if the…

Q: metric

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Q: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded.

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Q: Let X be a compact subset of R and let C(X) denote the set of continuous real-valued functions on X…

A: let X be a compact subset of ℝ and let CX denote the set of continuous real valued functions on X.…

Q: The set E={xER:xs2x² – 1} with the usual metric is compact.

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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: 5. State whether the following is true or false m) A subset I of a metric space R with the usual…

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Q: Given, we have a finite purely non-atomic measure space (X, F, µ), prove that µ(F) = [0 µ(X)].

A: Given: Here, the given is finite purely non-atomic measure space (X,F,μ). To prove that μ(F)=0, μ(X)

Q: if x is a compact Metric space and F e C (X₂Y) is equicontinuous, then F i's uniformly…

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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: (b) (i) Give an example of an infinite collection of nested open sets in R under the usual metric on…

A: Here given usual metric on the set of real numbers R

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.