If U is homeomorphic to V and U is compact, then * V is complete O V is connected V is compact V is not necessarily compact

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: Let X = [0, 1] U [2, 3] endowed with the standard metric. Then [0, 1] is an open subset of X. Select…

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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 (N, d) be the usual metric space, then the set A= {1,3} is O closed only O neither open nor…

A: ( N,d) be a usual metric space of natural number.  A={1,3} be a finite subset of N. Since A is…

Q: If there exists open ball B (a, r) such that B (a, r) NG= Ø then a is not limit point of G True O…

A: TRUE

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

A: In order for a set to be dense, it is required to be equipped in Tp and also as in the set 5 is not…

Q: Let (X, I) be a topological space and suppose that A and B are subsets of X such that ACB. Prove…

A: Let x ∈A°. Then by definition, there exists an open set U  such that  x∈U⊆A.Since A⊆B hence x∈U⊆B ,…

Q: 5. Let V be an inner product space over C. Show that for any v, w e V (v, (v, w)w) = | (v, w) |² and…

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Q: If there exists open ball B (a, r) such that B (a,r) nG = Ø then a is not limit point of G True…

A: TRUE

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

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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 the following statements: (i) If A is not compact then there are infinitely many open…

A: Consider the following statements: (i) If A is not compact then there are infinitely many open…

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: In order to be dense in R, R has to be equipped in the Tp. Also as in the interval, 5 is not…

Q: 5. If r = (r, (x, y, z) and ro = (xo, Yo, 20), describe precisely the set of all points (r, y, z)…

A: We plug the given values in the given equation and simplify

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: let (x, m) be a measurable space. then ¥8:% [, o] are measural ble funetion, then ) I +9 is mea…

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Q: Let X be a discrete spaces then O X is homeomorphic to R if and. only if X is countable O X is…

A: I have provided a solution in step2

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: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify tha the quotient…

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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: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

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Q: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

A: Given: A subset (a, b)⊆ℝ is T2-space in ℝ, Tu. To determine: Whether given statement is true or…

Q: (7) Which of the following is disconnected space: (a) (R\ {0}, Tusuat) (b) (R, Tusuat) (c) (X, t).…

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Q: Let ( Xr d be ame tric space,show that exist atopology on X induce b

A: Let (X, d) be a Metric Space We know that in a Metric Space  (1) arbitrary union of open sets is…

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: If S, S, are two subsets of an inner product space V(F) then show that

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Q: Prove that: 1. (R, |. |) is a complete metric space. 2. (Q.I-1) is not complete metric space.

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Q: Let A, B Be two subsets af R, such that B is bounded and YaE A ]bEB such that show that: a sb O A is…

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Q: If there exists open ball B (a, r) such that B (a, r) nG=Ø then a is not limit point of G

A: To check whether there exists open ball B(a, r) such that B(a, r) ∩ G= ∅, then whether a is not…

Q: Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact. * O…

A: "Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact." The…

Q: Let C(X,Y) have the compact-open topology. Show that if Y is Hausdorff, then C(X, Y) is Hausdorff

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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: 5. Let V be an inner product space over C. Show that for any v, w ɛ V ||u||² + ||v||² (v, (v, w)w) =…

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Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: 6) Let R be an equivalencerelakion CRis uni-Summitry them R=A

A: Given that R is an equivalence relation. If R is antisymmetric, then we have to prove that  R=D=x,…

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: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = A O (A)° = A – int A %3D…

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

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

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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: Suppose U and v are conformally equivalent. Prove that if U is simply connected, then so is V.…

A: Given that U and V are conformally equivalent that is there exist a mapping f from U to V such that…

Q: Assume X is Hausdorff. If K C X is compact and x ¢ K, show that there exist disjoint open sets U and…

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Q: In general, for a a) S is compact 2) Sis closed at S is compact

A: We use definition of compact metric space, totally bounded set, Closed set, accumulation point or…

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

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Q: 3. Let V be a finite dimensional inner product space. If S = {u, v} is an orthonor- mal set, compute…

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Q: Let E,F ⊆ X,d be compact subsets of a metric space X. Just using the definition of compactness…

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Q: Let X= (3, 5] u {6,7,...} and A= {5,6,7} and metric defined on X as d(x,y) = |x - y|

A: Note: As per bartleby instruction when more then three questions for subpart is given only three…

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

A: