Assume X is Hausdorff. If K C X is compact and x ¢ K, show that there exist disjoint open sets U and Vso that x E U and K C V.

Q: Exercise. Is it possible for a metric space (X, p) to contain more than one point and be such that…

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Q: Prove that in a metric space (S,d) every closed ball Sr[Xo] is a closed set

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Q: Let (X, d) be a metric space and let A ⊆ X be a compact subset. show that, If B ⊆ X and d(A, B) =…

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Q: For any subset A of a topological space X, we define the boundary of A to be: JA = AnX\A. a) Show…

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Q: Consider the set P = {f€C([0, 1], R) | ƒ is nondecreasing} Determine if (P,d2) is separable. Here,…

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Q: (b) Prove that the set S = {x EQ: √2<x<√3} is closed and bounded in (Q.1-1), but not compact in…

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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)…

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Q: Let x be O and all subsels f X whose complements Prove that (X, z) infimite set and let z consist of…

A: Consider, I=A⊂X : Ac is countable ∪ϕSince,Xc=ϕ is countable, X∈I Also given, ϕ∈I Assume, Aii∈N∈I…

Q: Consider R² with the usual topology, let A and B be two subsets of R2. We denote the closure of a…

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Q: What can you say about a set ACR, if every subset of A is closed in R? Which of the following…

A: Given: Every subset of A is closed in ℝ. To do: What conclusion can be given about A.   Conclusion:…

Q: Given the following relation on sets : R={(x,y)1x is a proper subset of y}. Which properties hold…

A: The given relation:R=x,y| x is a proper subset of y.

Q: Let

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Q: If M is a closed linear subs r space X, and x, be a vect the distance from x, to M, ional f, in X'…

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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: The set S = {T, (U) | U is open in X} U {T, (V ) | V is open in Y } is a subbasis for the product…

A: The given theorem is related with the topic product topology. Given that S = π1-1U | U is open in X…

Q: If CI(A) denotes the closure of A, which of the following is not correct in general for a set A: *…

A: cl(A) denotes closure of A

Q: Let (X,d) be any metric space, then the only open sets and closed sets are and X O True O False

A: Given statement is false because if we take descrete metric space

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…

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Q: If U is homeomorphic to V and U is complete, then" O V is compact O v is not necessarily complete O…

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Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: The objective is to choose the correct option: Let X be a discrete space then  a) X is homeomorphic…

Q: Prove the following sets are not compact by finding an open cover that does not have a finite…

A: a) Given-  A set A=−1,1. Explanation- Consider a set Xn=-2,1-1n for n=1,2.... Then clearly,…

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: Is the following statement True or False? Justify each answer. If S is a closed bounded subset of a…

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Q: For sets A and B, define A+B = {a + b: a € A, b E B}. (a) Prove that if A and B are open, then A+ B…

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Q: If A is a subset of a metric space X, then show that A = (x: each neighbourhood of x intersects A}

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

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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

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Q: Prove that the boundary points of an open interval (a, b) CR are the points a and b.

A: A point a∈ℝ is said to be boundary point of a set A⊂ℝ, if Ba,r∩A≠∅ and Ba,r∩Ac≠∅ for every…

Q: Let X be a set with more than one element. (X,t) is a topological space such tha Vxe X,{x} €T then…

A: Given a set X with more than one element.

Q: Let X be an uncountable set and let T = {U C X :U = Ø or U¢ is finite} %3D Then is topological space…

A: let X be an uncountable set and T=U⊑X : U=∅ or Uc is finite i.e cofinite topology on X. Then the…

Q: If U is homeomorphic to V and U is compact, then * O Vis compact V is complete V is connected O V is…

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Q: Let A, B be bounded subsets of R. Let A+B = {r+y:1 € A, y E B} and AB = {ry : 1 € A, y € B}. Which…

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Q: 4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show that E is closed.

A: Let (X,d) be a metric space, with x∈X , ε&gt;0, and E={y∈X:d(x,y)≤ε} To show: E is closed. That is…

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

A: Use the following definitions, to prove the result. If a set contains all its interior points, then…

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))…

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Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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

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Q: For any metric space (X, d) and x EX, the singleton set (x) is closed. Select one: OTrue O False

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Q: Prove that if X,d is a metric space and G ⊆ H ⊆ X, then G-bar ⊆ H-bar.

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Q: then: Suppose that f:[a,b]→R is bounded , P is parition of [a,b] and Q is refinement of P, L(f;Q) <…

A: It is a mcq type question and not asked about justification so i have provide direct the option.

Q: Let A be compact and U C A be closed. Prove that U is compact.

A: The complete proof is in given below

Q: Consider the following subsets of R equipped with the Euclidean topology: Зп — 1 A = {- n+1 B = Ü 1+…

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Q: f A∩B≠∅ and A∩Bc≠∅ then show that A∩X\B̅≠∅ and A∩B̊≠∅

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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 x1 and x2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: Let R be equipped with the standard topology. Define q : R? → R, x1, if x1 2 x2 X2, if x1 < x2…

A: See the attachment

Q: If A is a compact subset of a metric space (X, d) and B is a closed subset of A, prove that B is…

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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: Let A be a non-empty set of positive nunıbers, which is bounded above and let B = {x e R: 1/x e A}.…

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