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

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: QUESTION 8 The set M with the usual metric is connected. 2 O True O False

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

Q: (7 ifx # y to if x y Prove d(x, y) = is a metric on R (real numbers). (9 points) V.

A: …

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

A:

Q: By using the standard metric X. on Determine JA and X\A. Justify your Justi Py answer State whether…

A: Since you have asked multiple questions, we will solve the Question Q1 (i), (a) and (b) for you. If…

Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the Manhattan metric. Hint:…

A: Assume that A is open subset of R2  in an Euclidean metric. So A can be written as,…

Q: 16. The set S = { x∈R: x2 - 4<0} with the usual metric is

A: The set S = { x∈R: x2 - 4&lt;0} with the usual metric is ..

Q: 4. i) Let X = B [a, b] and let d be the sup metric on X. If n → x in (X,d) and an is continuous at…

A:

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: 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: Let M = 0,2) and d be the usunl metric, then dis not complete O True O False

A: False

Q: Let X = N and d be the usual metric, d (x, y) = |x – yl then d can induces a norm on X True O False…

A:

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

A:

Q: QUESTION 8 The set M={XEN:x<} with the usual metric is connected. O True O False

A:

Q: Suppose (X, d) is a metric space and let r> 0. Show that p: X2 → [0, c0) defined by p(x, y) = r·…

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: 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: d, d': xxx-R be two metrices on x that Satisfy following for every nEX and >0 7>0 st. Bai (x, S) C…

A:

Q: Suppose d and p are metrics on X: let Xa denote the set X with the topology induced by p; let f: Xa…

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

A:

Q: Consider A = [3,7), B = (7,9] as a subset of R with the usual metric  dR (x,y) = abs(x-y). Show Ac…

A:

Q: QUESTION 8 The set M= EN: X< with the usual metric is connected. O True O False

A: Since you have asked more than one question. So we will solve first one for you. If you want other…

Q: Define new metric which is different from giving in above questions on X in question 01.

A: With the help of questions 01 we define a new metric on X

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

A:

Q: let M= To,2] anel d be bhe usual metric then d is not compact or Compact ?

A: A set is said to be compact if and only if it is closed and bounded.

Q: d(x,y) 1 - Y -

A:

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

A: We have to decide whether the given statement is true or false.

Q: Draw the unit “circle” around the origin in the Euclidean metric, the taxicab metric, the hub…

A: The unit circle around the origin in Euclidean metric is The unit circle around the origin in…

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

A: We Know that a set is separable in metric space only if it contains a countable dense subset in it.…

Q: Given the set X = {x € R³ : d(x, a) . True False Rationale:

A:

Q: If X = {a, b} , how many metric can be de fined on X %3D infinite O none O only one O only finite

A: only one

Q: QUESTION 5 The set R/{O1 with the usual metric is not connected. O True O False

A: 5. TRUE The set, R - {0} with the usual metric can be written as the disjoint union of, (-infinity,…

Q: Let (R, d) be discrete metric space, then R is not separ able O True O False

A:

Q: Let X={9,10,11,..} and A= {10,11,12) and metric |defined on X as d(x,y) = |x-y Give the last answer

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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: Show that nonnegativity of a metric follows from (M2) to (M4).

A:

Q: Given the set X = {x € R³ : d(x, a) < r}, where d is the usual metric on R³. 3 The set X C R³ is an…

A: Solution

Q: Let X = R? , and let d2 be the usual, Euclidean metric on X. Define another metric, d : X x X → R,…

A:

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: Define on R, the metric d defined by d(x, y) = |1+1리 1+ |y| Then (R, d) is not complete. True False

A: Given

Q: 7 The set M={xEN:x<÷} with the usual metric is connected.

A:

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

A: Ans:

Q: Which of the following is not a cover of set of real numbers with the usual metric? O A. G={R O B.…

A:

Q: For any non empty set X, the discrete metro d:  X^2 to {0,1} is defined to by  d(x,y)={0, x=y and…

A:

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: Let d(x, y) = be a metric on R* then the open ball of center 2 and radius is: O ]1/2,1[ O None of…

A: I have used definition of open ball in metric space.

Q: Let X= (3, 5] u {6,7,...} and A= {5,6,7} and metric defined on X as d(x,y) = |x - y|

A: Note: As per bartleby instruction when more then three questions for subpart is given only three…

Q: Let f: (N,d) (N, d), where d is the usual metric and di is the discrete metric, defined f as f (x)…

A: