Q3. Let N = {1, 2, 3, ..} equipped with the cofinite topology. Prove that M = {7,14, 21} is a closed set.

Q: e following sets of points is collinear 2. L(-5,13), M(-10,10), N(-20,4)

A: Here we have to prove that each of the following sets of points is collinear.  L(-5,13); M(-10,10),…

Q: Is R, equipped with the finite-closed topology, Hausdorff? Provide a proof supporting your answer.

A: Given: ℝ is equipped with the finite-closed topology. Let, ℝ with this topology be Hausdorff and x,…

Q: 3.2 Prove that in any metric space (S, d) every closed ball S,xo is a closed set.

A: Let (S, d) be a metric space. Consider a closed ball Srx0 with both center x0∈S and radius r>0…

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: Let X be any infinite set with the co-finite topology. Identify which of the following separation…

A: Let X be any infinite set with co-finite topology. Definition:  Let X be any infinite set. The…

Q: Let T be the Euclidean topology. Let A = [0,2] and B = [0,1[, then B is clopen in (A, TA) where TA…

A:

Q: 29. Let N be with the co-finite topology If A={1,3,5.), then Bd(A)=

A:

Q: 2. Let X = {1,2, 3, 4} and let T = {ó, X, {1}, {3},{1,3}} be a topology on X, then {2, 4} is a. open…

A: Since you have asked multiple question , as per our guidelines we are supposed to answer only one…

Q: Let X = {1,2, 3, 4} and let T = {, X, {1}, {3},{1,3}} be a topology on X, then {2, 4} is a. open in…

A:

Q: a. Show that Ø, R E T. (by definition of Topology)

A:

Q: 1. Let X = {a,b,c} and B={{a,c}.{b,c}}cP(X). Show that B cannot be a base for any topology r on X.

A: According to the answering guidelines, we shall solve first question only. If you want others to be…

Q: * Let R be with the indiscrete topology. If A = {1,3, 5, 7, . }, then A' = A O N O RO

A: This question is related to topology, we will use the definition of indiscrete topology to solve it.

Q: 15. If t={X,Ø,{1,4},{4,3},{1,4,a},{b}} is a topology on X-{1,2,3,4}, then: * a=4, b=2 O a=3, b=1 O…

A:

Q: (a) Consider the following topology of X = {a,b, c, d, e}: T = {X, 0, {a}, {c, d}, {a, c, d}, {b, c,…

A: In topology and related branches of mathematics, a connected space is a topological space that…

Q: Which one of the following statements is true? * R with the Euclidean topology and R with the finite…

A: Answer

Q: 4) Consider the collection of intervals on the real line: B = { (a, b) | a 1}. Is this a basis for…

A:

Q: 2. Let X = {a, b, c, d, e, f}. Which of the following collections of subsets of X is a topology on…

A:

Q: of all Z € C that z iz e = e a Descnibe and sketch the set

A:

Q: Let X = {a, b, c, d, e} and7 = {X,6. {a}. {c, d}, {a, c, d}, {b, c, d, e}} be a topology on X, then…

A: Details solution given below.

Q: Let T cofinite topology on R and Y={5,6,7} find Ty?

A: We have to construct subspace topology with respect to the cofinite topology

Q: Let X = {a, b, c}, T1 = {X, p, {a}, {b}, {a,b}} and T2 = {X, p, {a}, {c}, {a,c}}. Then one of the…

A: Given that set X={a,b,c}

Q: Q2: Let the set under consideration be N for each n EN, define u, = {n, n+1, n+2, ...} and let 1 =…

A:

Q: 3 let X= {ab,c},B =} {n,b}, {9c3}.5how that Whether B is a %3D %3D base for any topology on X or not

A:

Q: Which one of the following statements is true? * O Rwith the Euclidean topology and R with the…

A: Which of the following statement is true?

Q: Which one of the following statements is true? O R with the Euclidean topology and R with the…

A:

Q: Q1.Consider X ={a,b, c, d, f, g} .Let T={ó, X, {a,b, c} ,{g};{a,b, c, g}} (a) Prove that T is a…

A: Since you post a question with multy part, we solve only first three, if you need any particular…

Q: Let lx, T) be a topological Space and A, B S X- if A is open, then show that An CIIB) s CLLANB). if…

A: Given X, τ be a topological space and A, B⊆X. Suppose, A is open then An arbitrary x∈A∩clB implies…

Q: Let X = {1,2, 3, 4} and let T {ø, X, {1}, {3},{1,3}} be a topology on X, then {2, 4} is a. open in…

A: Our objective is to choose the correct answer

Q: Which one of the following statements is true? * R with the Euclidean topology and R with the…

A:

Q: In a Cofinite topology of X the collection of subsets is: a) r = {AcX|A±Ø or A is finite} b) r = {Ac…

A: Let X be a non empty set , then the collection of subsets of X whose compliments are finite along…

Q: If X = {a,b, c}, let T1 = {0, X, {a}, {a,b}} and T2 = {0, X, {a}, {b, c}} . Find the smallest…

A:

Q: * Let R be with the co-finite topology. If A = {1,3,5, 7,..}, then Aº R O Q O A O N O

A:

Q: Q3. Let N = {1, 2, 3, ...} equipped with the cofinite topology. Prove that M = {7,14, 21} is a…

A: Given below the detailed solution

Q: 4) Determine whether or not each of the following subset of R with the euclidean topology is…

A:

Q: 1. Let T and S be topologies in X. Show that SnT is also a topology in X.

A: 1.

Q: Let Z be with the co-finite topology If A = {1,3,5, 7,.. }, then A A O O O

A: As Z is with co-finite topology. Every open neighborhood of an infinite set must contain infinitely…

Q: (ii) Find a connected subset of R² which is not convex.

A: Note: Hi! Thank you for your question. As per the guideline we can solve one question at a time in…

Q: Q. 1: If X = (a, b, c, d, e) Determine whether or not each of the following collections of subsets…

A: Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X: 1) Both the…

Q: 4) Determine whether or not each of the following subset of R with the euclidean topology is…

A:

Q: EX LeT X = {!,2,3,415,6} T- Ex,¢,てり,E4,しろ4), [ス456 Topology.?

A:

Q: 2. Let T be the cofinite topology on R, and let A = (-x, 1) U (1, ), B = (1,2). Fine the boundary…

A:

Q: R is not connected if T is * the indiscrete topology the trivial (usual) topology the finite closed…

A: Option 3 is correct.

Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: Let T be the Euclidean topology. Let A = [0,2] and B = [0,1[, then B is clopen in (A,T) where T is…

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

Q: 2. Let X = {1,2, 3, 4} and let T = {ø, X, {1}, {3}, {1,3}} be a topology on X, then {2, 4} is a.…

A: b. closed in (X,T)

Q: Ris not connected if T is * O the indiscrete topology O None of the choices O the trivial (usual)…

A: Every indiscrete space is connected.  as the union of two disjoint non-empty sets, so R is a…

Q: Let N be with the discrete topology If A=(1,3,5..), then Bd(A)=

A:

Q: Let X = {a, b, c}, T1 = {X, 0, {a}, {b}, {a,b}} and T2 = {X, 6, {a}, {c}, {a,c}}. Then one of the…

A: According to experts guidelines of bartleby i have to solve only first problem so repost for further…

Q: 1/ For any. topological space (X, T), if T has only three elements then (X, T) is connected.

A: Given: Topological space X, T and T has 3 elements  To prove that X, T is connected when T has 3…

Q: We define the included point topology by Tp={ UcR;U=Ø or peU}. Let A = [3,5[, then A is dense in R…

A: