) Prove the theorem that every metric space has a completion

Q: Exercise. Is it possible for a metric space (X, p) to contain more than one point and be such that…

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Q: Is the euclidean space R³ is a complete metric space

A: We have to check whether euclidean space R³ is a complete metric space or not .

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: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

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Q: are closed.

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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: 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: Every metric space is a hausdorff space .

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Q: . Prove or disprove every metric space is normed space,

A: A normed vector space or normed space is a vector space over the real or complex numbers, on which a…

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: 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: 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: Any cover of a metric space is open.

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

Q: If X is a metric space with induced topology Ƭ, then (X,Ƭ) is Hausdorff. The contrapositive of this…

A: Given : (1) (ℝ, T) with the topology induced by taxicab metric.              (2) (ℝ, T) with the…

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: (a) Prove that the metric space (2,11) is complete. (b) Show that the metric space (Q.H) is not…

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Q: (a) Prove that the metric space (2,11) is complete,

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

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

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Q: Theorem 2.25. Suppose that (X, p) is a metric space. (a) The intersection of any collection of…

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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: (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: Exercise. Prove that in any metric space (X, p), a closed ball {r E X : p(a, x) <r} is closed.

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

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Q: Define the Cartesian product of sets X and Y? How is this Cartesian product denoted?

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

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

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Q: Consider the spaceZ+with the finite complement topology. Is thisspace a Hausdorff space? Is this…

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

A: This is a problem of Functional Analysis.

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

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Q: prove that any finite dimensional normed space is reflexive.

A: Let us suppose that we have a finite-dimensional normed space, X For any natural number, n, let…

Q: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

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

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

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Q: Remark: If (V,) is an inner product space. Then for all uiv,we Vand a,p EK い = &+p リ = す

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Q: The closure of a compact subject of a regular space is compact.

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Q: Prove that Let X be a metric space such that X E1 U E2. If E1, E2 are complete, then X is complete.

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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: 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: 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: Theorem 9.8. A metric space is Hausdorff, regular, and normal.

A: The objective is to show that every metric is hausdorff, regular and normal. Let x≠y be points of a…

Q: Exercise. Prove that in any metric space (X, p), a closed ball {x E X : p(a, x) <r} is closed.

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Q: Remark: If (V, = « +p vi " = a + p

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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