Q. Assume that (R,T) is the usual Euclidean topological space. Let E = {0, ±2,+4, +6, ..} CR be the set of even integers.Then а. dE - ф аnd ӘE - E. b. dE = E and DE = 0. с. dE d. dE E and DE = E. ф and ӘE — ф. || a b d

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: Ex: Let (X, d) be a metric space and E, andE2 EX Show that whether each one of the following…

A: According to guidelines I solve first 3 subparts so kindly request you to repost the remaining…

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A: 3.3 is also known as Hausdorff Property. d(x1, x2) > 0 for every x1 ≠ x2   (metric space…

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

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

Q: Q. Assume that (R, T) is the usual Euclidean topological space. D= {(x, y) E R?, y = x² +3} C R.Then…

A: Solution: Here the subset of ℝ2 is D=x,y∈ℝ2,  y=x2+3 Now, The set D contains points x,y which…

Q: Let A and B be subsets of topological spaces X and Y, respectively. Prove that CI(A × B) Cl(A) ×…

A:

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: 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 (X, I) be a topological space and suppose that A and B are subsets of X such that ACB. Prove…

A: Let x ∈A°. Then by definition, there exists an open set U  such that  x∈U⊆A.Since A⊆B hence x∈U⊆B ,…

Q: Let (x,d) be metric space and A, BC X. 2. Show that (AnB)= A'n B and AUB = AUB %3D

A:

Q: let and be nonempty metric Space x E X. Ihen A B Sebsets of and let d(x, A), dlA, B) and dlA) are…

A: As per bartleby guidelines for more than three subparts in a question only first three should be…

Q: Q7. Assume that (R,T) is the usual Euclidean topological space. N = {1,2, 3, ....} C R. Determine…

A: There is no interior point of N.

Q: Let (X,t) be a topological space, A, B C X. Then A' UB' = (A U B)', A'n B' c (A N B)' Select one: O…

A:

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: Let A be a non-empty subset of the metric space (X,d). show that |d(x, A)- d(y, A)| < d(x, y)For all…

A:

Q: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

A: Given: A subset (a, b)⊆ℝ is T2-space in ℝ, Tu. To determine: Whether given statement is true or…

Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

Q: 2. Let (X, d) be a metric space and y E X. (i) Prove that the closed ball B[y,2] is a closed set in…

A:

Q: Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

A:

Q: In a metric space (X,p), if lim un = u, lim v, = v, show that lim d(un, Vn) = d(u, v). %3! n-00

A: Convergent Sequence: A sequence xn in a metric space X, d is said to converge to a point x∈X if for…

Q: Let (X, d) be metric and PXIR space, dosy) I+ dexy) a de fined by qusN pusy) = ニ show that p is…

A:

Q: 3. Find the closure and interior of the set A = {(x, y): xy = 1} as a subset of R? (equipped with…

A: Closure and interior

Q: Q # 3(b) Let (G, H) be a disconnection of a topological space (X, 7), ant let AC X. If A is…

A: Before going to this proof, we prove the below statement which in turn proves the above result.…

Q: Which of the following are compact spaces? (a) S" with a finite number of points removed. (b) The…

A:

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.…

A:

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. biso (2) If…

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

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

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

A:

Q: Let 1={Ø}U{UER:NC U}, -n+3 ). A = :n EN and B = [0, ∞). Then n+1 in the topological space (X, 1) O…

A:

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

A:

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

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 (S, d) be a metric space and suppose that ρ:S×S→R is defined by ρ(x, y) =d(x, y)/(1 +d(x, y))…

A:

Q: Q. Assume that (R,T) is the usual Euclidean topological space. Let E = {0, +2,+4, +6, ..} CR be the…

A:

Q: d(r, y) Let (X, d) be a metric space and let p(r, y) = for all r, y E X. Show that (X, d) 1+d(x, y)…

A: According to our company policy we are supposed to answer only first question i.e. (4) 4) in this…

Q: Q7. Assume that (R, T) is the usual Euclidean topological space. N = {1,2,3,..}c R. Determine N°,…

A: ANSWER :

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

A: Ans:

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: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.

Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

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

A:

Q: (c) If X={1, 2, 3}. T= {4, X, {1}, {1,2}, {3}, {1,3}}. JUSTIFY whether the topological space (X, t)…

A:

Q: c) Let (M, d) be a metric space and let S,T≤ M. Prove that SOT = ST. But SOT SOT.

A:

Q: Let X = {a, b, c} Define d: X2[0, 0) by d(a, a) = 0, d(b, b) = 0, d(c, c) = 0, d(a, b) = 1, d(a, c)…

A: Metric Space: Let X be a set and a function d :X×X→ℝ Then X, d is a metric space if d satisfies the…

Q: Prove the following: Let (Y, d) be a metric space. Let B be a bounded subset of Y. If A C B, then A…

A: Let Y, d be a metric space and B be a bounded subset of Y. Suppose A⊂B. To prove that A is bounded…

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

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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: Show that the real number R with d(x,y)=|x-2y| are not a metric space

A:

Q: A° U B° c (A U B)

A: