State whether each of the following statements is true or false, in either case substantiate/justify. (e) Every connected metric spaces is compact. (f) The set B = { (x; y) ∈ R2 : x2 + y2 < 1} as a subset of R2 with indiscrete metric is disconnected. (g) The empty set is path-connected.

Q: Let M = {0,1} and d be the discrete metric, then d is not complete %3D O True O False

A: Discrete metric space.

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

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

Q: (1) Determine if each of the following statements is true(T) or false(F). 1- In a metric space,…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

Q: Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B)…

A:

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: A subset I of a metric space R with the usual metric is compact if and if only it is an interval…

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

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Q: On a previous homework, you proved a Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2,…

A:

Q: Let (X, d) be a separable metric space. into l∞.

A: This is a problem of functional analysis.

Q: Every T3 space is regular space. True False

A: Since a regular T1 space is called a T3 space.

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

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Q: Let X = R, and Let M = [0, 3] and d be the discrete metric, then M is

A: M=[0,3] is compact ,since it is closed and bounded in real R.

Q: On a previous homework, you proved a Bolzano-Weirstrass theorem for R3 with the metric d((x1, X2,…

A: Given a metric in ℝ3 and given it satisfy Bolzano-Weirstrass theorem. We need to prove that if a…

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

A:

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

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Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

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

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

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Q: State whether each of the following statements is true or false, in either case…

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Q: The set [0,4] with the usual metric is sequentially compact. O True O False

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Q: Let (X, d) be a metric space and B = S(xo,r) be the open ball with centre at xo E X and radius r> 0.…

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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: Let X = R, and d be the discrete metric, and A = [3, 6) Show that if A is compact or not

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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: (1) Determine if each of the following statements is true(T) or false(F). 1- In a metric space,…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

Q: Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d generate the…

A: Let X, d  be any metric space. It is required to prove that there exists a metric d such that d and…

Q: The set S=xER:x²-4<0} with the usual metric is . O A. Compact. B. connected. C. Not connected D.…

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Q: For every ,x, y E R, let p(x, y) =| x² – y² |. Is p a metric -

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Q: Give the definition of a separable metric space (S, d). Show that the metric space (R, d), where…

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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 Completion Theorem below to show that the Completion of the discrete metric space X is…

A: By completion theorem, there exists a complete metric space which has a subspace that is isometric…

Q: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

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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: 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: Given the family C[0, 1] of continuous functions ƒ [0, 1] → R, and the norm ||f|| = sup |f(x)]. x=…

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Q: For the following subsets in an indicated metric space determine the interior, boundary points and…

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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: Show that (C(X),|| · ||∞,x) is a metric space.

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Q: S={xER:x2 - 4<0} with the usual metric is A. Compact. B. connected. C. Not connected D. Sequentially…

A: Given set  S= {x€R : x2 -4&lt;0}

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

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Q: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

Q: 2. Does d(x, y)= (x-y)? define a metric on the set of all real numbers?

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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: (a) Prove that the metric space (Z,) is complete.

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

Q: Give the definition of a separable metric space (S, d). Show that the metric space (IR, d), where…

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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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Q: (b) Describe graphically the set A={(x,y):d[(x,y), (0,0)]<3}for the following metric space on R²…

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Q: Prove that: 1. (R, J. |) is a complete metric space. 2. (Q, I. |) is not complete metric

A: To prove that R,. is a complete metric space.  Suppose that xn is increasing and bounded, Let…

Q: Prove that any smooth manifold (M, A) has a countable smooth atlas V (so that D(V) = A)

A: