Is the set S = [0,1] with the discrete metric d separable? Explain

Q: Exercise. Show that in R with the euclidean metric, any subset of R\Q with more than one element is…

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Q: Let (R. d) be discrete metric space, then Ris separ able O True O False

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Q: For the metric space (R, d) with d the usual Euclidean metric, which sets are in the topology T…

A: Wr Know that,"Let X = (-∞, ∞) and C consists of all intervals of the form (a,b).The arbitary union…

Q: If X =R and d(x.y)30, then X metric space not metric semi metric

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Q: The taxicab metric p on the plane R² is defined by the rule: p((x1, y1), (x2, Y2)) = |x1 – x2| + |y1…

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Q: State whether the following is true or false: a) The set [0,4] with the usual metric is…

A: (a) The set [0,4] with the usual metric is sequentially compact. This statement is true. Considering…

Q: Let X = R. and d is discrete metric, Find S (2,2) OR O (2) O {0} (2)

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Q: Exercise 4. If d is a metric on the set X, we define dcut : X × X → [0, 1] by dcut (x, y) = min(d(x,…

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Q: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

A: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

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: Let d be a metric on X + 0. Show that the function e defined by d(a, b) 1+ d(a, b) e(a, b) = %3D…

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Q: Define p: R^2 x R^2 to [0, infinity) by the rule p(x1,y1),(x2,y2)=squareroot((x1-x2)^2+(y1-y2)^2)…

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Q: Show that d(x y) = √x-y defines a metric on the set of all real numbers

A: We will use the basic knowledge of metric spaces and set theory to answer this question.

Q: Let X=R2 and let d be the Euclidean metric.  Define d3:  R2 x R2 to R by d3(x, y) = min{1, d(x, y)}…

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Q: Let p1 and p2 be points in a metric space X. Prove that the set p1 ⋃ p2 is compact in X. Now, let…

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Q: Verify that the discrete metric is actually a metric.

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Q: Let (X, d) be a metric space, and p: X→ IR doxy) I+desy) de fined by puM pusy) = %3D show that p is…

A: Consider a set X and a function defined from it to the real numbers as d : X×X→ℝ. Then, X,d is a…

Q: Set (Q, M, µ) a space of o-finite measure. we define the aplication 4: L° (Q) → L'(Q)* given by =…

A: Let g,h∈L∞(Ω and α∈F Claim:- ϕ(αg+h)=αφ(g)+φ(h)

Q: (f) The set B = {(x, y) E R² : x² + y² < 1} as a subset of R? with indiscrete metric is…

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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: b) Let X = R equipped with the standard metric. Then The boundary of [a, b),(a,b),[a,b), and(a, b)…

A: This is a problem of topology.

Q: 1. Let, y eR. Determine if the following are metrics on R or not (a) d, (x, y) = /1x - y| (b)…

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

Q: Exercise. a) For every ,r, y E R, let p(r, y) =| x² – y? |. Is pa metric U..... on R? b) For every…

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

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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: (f) The set B = {(z, y) E R² : ² + y < 1} ; is disconnected. as a subset of R? with indiscrete…

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Q: Find all limit points of a set A = {(2+,2 – 1)|n e N} on the Euclidean plane with the usual metric.…

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Q: Exercise.3 Show that the following the functions d and p on X = R? are metrics or not, such that 2.…

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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: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y+ z) = d(x,y), x,y € N…

A: Answer

Q: . Let X be a nonempty set with metric d. Show that the function e defined by e(x, y) = min[1, d(x,…

A: We know that d(x,y) is a metric on X. Hence d satisfies all the properties of a metric.  Now e(x,y)…

Q: Let a = (a, az); b = (b, ,b2) € R?, define the function %3D %3D d: R? x R? ---> R by d=(a, b) =…

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Q: d: R? x R? R Q2: Let d(a, b) = |b, - al + do(az, b2), where do denotes the discrete metric on R…

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Q: Given the discrete metric space for n E Na the discrete metric o defined by the distance function 0…

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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: If (X, d) is any metric space, show that another metric on X is defined by d(ry) d(r, y) 1+ d(x, y)…

A: The given problem is to show that X is bounded in new metric d(x,y)  Already given (X,d) is the…

Q: If X =R and d(x,y)= 1, then X not metric semi metric metric space

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Q: Define a metric as d(x,y)=|x1-y1|+|x2-y2|. Find the set of all points x=(x1,x2) in R2 such that…

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Q: Let (X, d) be a metric space. Define dˆ : X × X → R, by: ˆ
d(x, y) = min{1, d(x, y)}. (a) Prove that…

A: To verify that d^ satisfies the axioms of a distance function

Q: Let f function from (R,d) to (R,d1) defined as d(x) = x, where d is the usual metric and .di is the…

A: Metric space function

Q: Show that the Euclidean metric is a metric( i.e. verify that the function d satisfies the three…

A: Here we have to show thatt Euclidean metric is a metric.

Q: 9 on X= IR, k>o. Define metrics ond q P is not in the form kq for

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Q: EC. Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric). i. Explain…

A: We will use facts from general topology to solve this question.   Note that we know that [a,b] and…

Q: (a) Consider the metric space 0 (i) if r 1.

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

Q: Let A be a subset of R with the usual metric such that A=A. If there exists r>0 such that d(x,y) <r…

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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: (X,d) be a metric space. Define f: X x X -> Real Numbers by f(x,y) = d(x,y) / 1+d(x,y) . show that f…

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Q: Let X = R², x, y EX where x = (x₁, x2), y = (y₁.Y2) with the metric function (4 pts) (x, y) =…

A: This is a question of topology and metric space.

Q: Exercise 1.33. Let (X,d) be a metric spáce and d(x,y) 1+d(x,y)' d* (x,y) = Prove that the family of…

A: This is problem from my old works, where I used e instead of d*.