2- Moke the definitions of open and closed sets. x =Larbic,d,eļ and ノ closed subsets of (x,T) space, beth open and closed subsets, Identify neither open nor closed subsetsropen but not closed subsets, and closed but not open subsets.

Q: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

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Q: 28. Let R be a collection of sets defined by R = {x | x ¢ x}. %3D (a) Show R is not the empty set.…

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F open?

A: A set is open if and only if it contains all of its limit points. Also recall that the union of open…

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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A: The "finite closed topology" describes the closed sets. A subset U is open if and only if U=∅ or X∖U…

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Q: 18 What is the size of the space stretched by the set {x, x+x², x³, 1 , x²} ?    a) 2nd  B) 34…

A: Solve the following problem

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A: Let X be a topological space then, To prove: Following statements are equivalent: (i) X is…

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Q: (ii). Prove that a set S CR is everywhere dense iff it intersects every non-empty open sets in…

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

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Q: 5. Show that if A and B are bounded subsets of R, then A U B is a bounded set. Show that sup(A U B)…

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

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

A: Let {Fk}  be an arbitrary collection of closed sets . . 1  i.e. for k = 1, 2, 3, . . . . . . . . To…

Q: al let (xt) betopology space and BCY then B is closed in y iff there is closed set CinX st B-COY

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Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

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Q: Theorem 2.1. Let{U}{_, be a finite collection of open sets in a topological space (X,T). Then N, U…

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Q: 7. Let X be an infinite set equipped with the finite closed topology. A finite subset of X is a.…

A: Option (a) is correct.

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Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

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Q: (iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the…

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

A: The given is related to Metric spaces. The solution is explained below.

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A: Topology Question

Q: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

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Q: Recall that RK denotes the real line with the K-topology. Let Y denote the quotient space obtained…

A: Given:-             Note that if U opens in R, and then U-K is opens in R k.             U = union…

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A: Given the set X contains more than one element.

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Q: 5. Let (X,T) be a discrete space, then Va E X, the set {a} is а. орen b. closed c. clopen d. is…

A: Our objective is to choose the correct choice

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A: X is finite

Q: 7. Suppose that a set X is equipped with the finite closed topology. Which one of the followings is…

A: As per the policy, we are solving first question. Please repost the question and specify which…

Q: Let X be an infinite set with the countable closed topology T={S subset of X :X_S is countable}.…

A: We can show that X is countable. And by taking singleton we can solve the problem

Q: space. connected subsets of X such that C; is non-empty, then C = C; is also

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Q: 2- In a metric space (X, d), A is the intersection of all closed subsets of X which contains A.

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

A: Let, s be the circle given by, s=x,y∈ℝ2 :x2+y2=2  Then s is a closed subset of ℝ2. Let us consider a…

Q: (2.2) Let C = {a, b, c, d} and T = {0, {a, b, c} , {b, c} , {a},C} be the topology defined on C. %3D…

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Q: Exercise 60 Let p and q be two points in a topological space X. Show that there exists an open set U…

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?

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Q: 5. Let (X,7) be a discrete space, then Va E X, the set (a} is a. open b. closed c. clopen d. is…

A: As per our guidelines we are suppose to answer onlyone question, kindly post the other questions as…

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

A:

Q: Let {Fa}aeA be a collection of closed subsets of R. (a) Prove, from the definition of closed, that…

A: Given that Fαα∈A is the collection of closed subsets of ℝ. The objective is: To prove that ∩α∈AFαis…

Q: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then* OX…

A: Gievn that Let T be the finite closed topology and be also the discrete topology on a non empty set…