Question 6. In a topological space (X,7) a subset ACX is called regular open if A=int(cl(A)) "A= A". Determine whether the family of all regular open sets is closed under arbitrary union or not.

Q: Question 3. Let , and r, be two topologies on X such that ,cr;. Construct a space on which a r-limit…

A: Given below the detailed solution

Q: Question 9. Prove or disprove. If (X,7,) and (X,r,) are T,-spaces, then (X,7,nt,) is also a T -…

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Q: Let A = {0, 1} with the discrete topology and let f: A R be defined by f (0) = -1, f(1) = 1.

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Q: 4. Let (X, T) be the topological space defined in question 2. Let A = {1,2,3}, then the set of limit…

A: Given problem is :

Q: Every closed and bounded subset of X is compa The subset {z EX: |||| ≤ 1} of X is compact... X is…

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Q: Problem 2. Let X be a countable set and let s e X. Consider the particular point topology Tp on X,…

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Q: Theorem 3.15. Let (X,T) be a topological space, and let S be a collection of subsets of X. Then S is…

A: Given  X,τ is a topological space and S is a collection of subsets of X. Explanation let S is a…

Q: Question 5. In a topological space (X,7) a subset ACX is called regular open if A=int(cl(A)) "A= A".…

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Q: (8) Show that the map f(z) = 1/z is a bijective holomorphic map from S? to itself, using the charts…

A: Note: A function is holomorphic if it is analytic everywhere except one point. Given: fz=1z, z≠0

Q: Theorem 3.7: A subset A of a metric space (X, d) is closed if and only if/.contains all its limit…

A: A subset A of a metric space (X, d) is closed iff A contains all its limit points.

Q: Question 7. Let A=[0,1]U(1,2)ng) be a subset in the usual topological space where Q is the set of…

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

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

Q: Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset of X such that…

A: Let C be a connected subset of the topological space X.Also given that D is a subset of X such that…

Q: Theorem 2.28. Let A be a subset of a topological space X. Then Int(A), Bd(A), and Int(X – A) are…

A: Given that A is a subset of the topological space X. We have to prove that IntA, BdA and IntX-A are…

Q: Theorem 6.3. If X is a compact space, then every infinite subset of X has a limit point.

A: We have given that, X is Compact space.

Q: on R. Question 3. Let r, and be two topologies on X such that Sr,. Construct a space on which a…

A: According to our guideline we can answer only first question only or three subparts  and rest can be…

Q: Question 5. In a topological space (X,t) a subset ACX is called regular open if A=int(cl(A)) "A A".…

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Q: Question 6. In a topological space (X.r) a subset Bof X is called a 7- open subset of X if…

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Q: Question 2 2.1) Let X = R and ( be a collection given by ( = {0}U{0cX:0°<}. (a) Show that ( forms a…

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Q: 6. Consider the horizonal strip 2 = {z+ iy : -1 < y < 1}. There exists a holomorphic function f on…

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Q: Question 3. Show your working in each case: (a) Find all the cluster points of the set M {m +: m, n…

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Q: Theorem 7.7. LetX be a space with a dense set D, and let Y be Hausdorff. Let f : X → Y and g : X → Y…

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Q: Question 2. space is the union of a finite number of closed balls radius e. Prove that a metric…

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Q: Question 9. Prove or disprove. If (X,7,) and (X,r,) are T, -spaces, then (X,7,nt,) is also a T, -…

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Q: Question 2. Let t be the topology on R consisting of R, 6 and all open infinite intervals E = (a,0)…

A: Since we know interior of any interval is open interval .   Here we have to given [5,8) Hence…

Q: Theorem 2.3. A set U is open in a topological space (X,J) if and only if for every point x€ U, there…

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Q: Question 9. Let A=(101,102,103., n} be a subset in the usual topological space where n is an…

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Q: Question 1. Let (X,r) be a topological space and let A be a subset of X. Determine whether the…

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Q: Question 1. Given a metric space . (a) If 2, y E X and p(x, y) 0, prove that a y. (b) Prove that a…

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Q: Question 5. Let S be a subbase for a topology r on X and let A be a subset of X. Show that the class…

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

Q: Question 3. Show your working in each case: (a) Find all the cluster points of the set M = {÷ +: m,n…

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Q: Question 1. Let X and Y be infinite sets. Let X be equipped with the discrete topology, and Y…

A: Given, X and Y be two infinite sets in topological space.  X is a discrete topology and Y is a…

Q: Question 6. Let (X,r) be a topological space and A, BCX. Determine whether (A-B)CA -B or not.

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Q: Suppose the collection = {B₁ (x): xe (0,1)}. Is B an open cover of the interval [0, 1) CR? Give an…

A: Given: The collection B=B12x : x∈[0, 1). To determine: Whether B is an open covering of the interval…

Q: Question 2 Prove directly (i.e. from the definition of compactness) that if K is a com- pact subset…

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Q: Question 4. Suppose that X = {x₁,x₂,.,x} is a subset Inner Product space. Which option is correct?…

A: We use definition of Orthogonal and Orthonormal set.

Q: Question (6): verse is not true. Contractible space simply connected space. But con-

A: A manifold X is Contactible if its identity map is homotopic to some constant map.   A manifold X…

Q: Problem 5. Let X be a topological space; let A be a subset of X. Suppose that for each I E A there…

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Q: Question 2. Prove that a nonempty closed subset of R, if it is bounded from below, has a least…

A: The given question is related with real analysis. We have to prove that a nonempty closed subset of…

Q: Question 2* Which one of the followings is compact a. A discrete space with at least to points. b.…

A: Given: Statements: (a) A discrete space with at least two points is compact. (b) Any set with the…

Q: Question 6. In a topological space (X,r) a subset ACX is called regular open if A=int(cl (A)) "A=…

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Q: Question 3 Show that a subset G of R" is open iff it is the union of a countable collection of open…

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Q: Question 9. Let A (101,102,103., n} be a subset in the usual topological space where n is an…

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Q: Theorem 2.20. For any set A in a topological space X, the closure of A equals the inter- section of…

A: Let A  be a set in the topological space X. To prove that the closure of A  is the intersection of…

Q: Question 8. Let A=[0,1]U(1,2)ng) be a subset in the usual topological space where Q is the set of…

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

Q: Question 8. Let A = {10,11,12} be a subset in the usual topological space. Find A'. Question 9.…

A: Since multiple questions are given in the question, the solution to the first question is given…

Q: Question 6. In a topological space (X,t) a subset B of X is called a y-open subset of X if BC(B°UB).…

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Q: Question 9. Prove or disprove. If (X,r,) and (X,r,) are T,-spaces, then (X,7,nt,) is also a T,-…

A: -space: A topological space is -space if it satisfies the following axiom of Frechet : "If x and y…

Q: Question 2. Letr be the class of subsets of R consisting of R, ¢ and all open infinite intervals E,…

A: Given below the detailed solution

Q: Question. 3 {x₁,x₂,.,x} is a subset of an Inner Product space. Suppose that X = a. If X is an…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.