Question 9. Let A (101,102,103., n} be a subset in the usual topological space where n is an arbitrary integer. Find A'.

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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

A: We have to prove that A is Closed if and only if A contains all its limit points. Note : In proof i…

Q: 2.4.5. Let X be a metric space. Given x E X, prove that {x} is a closed subset of X.

A: Let (X,d) be a metric space to show that {x} is closed sunset of X.

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

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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…

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Q: Question 7. Let (X,r) be a topological space and A, BCX, Determine whether A-B2A-B or not.

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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: 3.1 Show that an infinite intersection of closed sets E k = 1,2,3,., in a metric space (SA) is a…

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Q: Let ([0,1), d) is a met ric space. The set G= [0, is open in (R,d) .A O (open in ([0,1),d.B O (not…

A: Option (B) is correct.

Q: - Let (X, t) be a topological space and A, BCX, then 1- A° 2 A- b(A) 2- A° # A- b(A) 3- A° = A –…

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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: 3.2 Prove that in any metric space (S.) every closed ball Se[xo] is a closed set.

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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: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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

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Q: Theorem 2.26. Let A be a subset of a topological space X. Then p is an interior point of A if and…

A: First well show that, if there existsan open set U with P∈UCA, thenP will be an interior point of…

Q: 7. Let X = {a,b,c}. Show that Bß ={{a,b},{b,c}}cannot be a base for any topology on Х.

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Q: 3. Let (X, T) be the topological space defined in question 2., then the closure of {2} is a. X b.…

A: It depends on the given topological space. Here topological space is not specified. If it's usual…

Q: 4. Let X be any set and 7 be the countable complement topology on X. b) If X is uncountable, then…

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Q: Problem 6. Give an example of a metric space (X, d) and a (necessarily infinite) family of open sets…

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Q: Theorem 3.16. Suppose X is a set and S is a collection of subsets of X. Then S is a subbasis for…

A: Given that X is a set and S is a collection of subsets of X. Let B be the family of finite…

Q: Question 10. Prove or disprove. Any infinite subset A of a discrete topological space (X,r) is…

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Q: Question 1. Let (X,7) be a topological space and let F be the collection of all closed sets, Show…

A: Closed Sets: A subset of a topological space is said to be closed if the set is open De Morgan's…

Q: QUESTION 9 Suppose metric space X is not path-wise connected then X is not connected. True False

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Q: Problem 3. Let X be a finite set. Prove that the finite complement topology on X is the discrete…

A: Note: The finite complement topology is also called cofinite topology. And the discrete topology is…

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 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: Question 1. Let A be a subset of a topological space (X,7) with the property that each point pEA…

A: Given below the detailed solution.

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: Let (N, d) be the discrete metric space, then the set A = {2,3, 4} is closed but not open open but…

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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: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: Theorem 2.15. Let (X,T) be a topological space, and let U be an open set and A a closed subset of X.…

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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: 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: 3.6 Let A be a subset of a metric space (S, d). Show that if H is a nonempty open subset of S and An…

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Q: Question 6. In a topological space (X,7) a subset ACX is called regular open if A=int(cl(A)) "A= A".…

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

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: 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: Question 7. Let (x,r) topological space and A, BcX. Determine be a whether A-B2A-B or not.

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Q: Question 1. Let (X,r) be a topological space and let F be the collection of all closed sets, Show…

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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: 3.1 Show that an infinite intersection of closed sets F, k 1,2, 3,..., in a metric snace (S, d) is a…

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

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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: 3.3 Let 11 and ra be distinct points in the metric space (S, d). Verify that there are open balls…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.