Let (X,τ) is a topological space and A ⊆ X. If all subsets of A are closed in X, then set A cannot have a limit point.

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: Let A and B be subsets of topological spaces X and Y, respectively. Prove that CI(A × B) Cl(A) ×…

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Q: Prove that a point x is a limit point of a set A if and only if every e-neighborhood of x intersects…

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Q: Let (X, d) be a metric space with X being an infinite countable set. Show that X is not connected.

A: A set is said to be infinite countable if its elements can be counted like natural number but you…

Q: The only subsets of R which are both open and closed under the discrete metric are Ø and R. Select…

A: This is False statement, in discreate metric space, every subset of R is both open as well as…

Q: If F is a closed bounded set, then every infinite subset S of F has a limit point in F.

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Q: Let X be any set with the discrete topology. Show that X is normal.

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Q: Let (X, d) be a metric space with X being a countable set. Show that X is not connected.

A: Let (X, d) be a metric space with X being a countable set. Show that X is not connected.

Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

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Q: Suppose that X and Y are topological spaces and f: X- Y. Show that f is continuous iff for each…

A: Definition: Let X and Y are topological spaces and f : X→Y be a function. We say that the function f…

Q: A map f:x-y is said to be an open map if for every open set U of X, the set f(u) is open in Y. Show…

A: Given, a map f: X→Y is said to be an open map if for every open set U of X, the set f(U) is open in…

Q: Let A be any set with the discrete topology. Show that A is normal.

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Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Two discrete spaces are homeomorphic if there exists a bijection between them and it is possible if…

Q: Let A be a subset of a topological space X. Show that bd(A) = Ø iff A is open and closed in X.

A: We need to prove the statement.

Q: Let (X, d) be a separable metric space. into l∞.

A: This is a problem of functional analysis.

Q: If S is a closed bounded subset of a metric space X, then S is compact.

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Q: Let

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Q: If X is a normed space then the open ball and closed ball is a Example: convex set

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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: Let A be a subset of a space X and let x ∈ X. Then x ∈ A iff there exists a net in A (that is a net…

A: Given: Let us consider the given, A be a subset of a space X and take x ∈ X.

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

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Q: Let X = {a,b,c} prove that topological space on the set X T = {X. Ø, {b}, {a,b}} be a

A: Let X={a,b,c} be a set. To show that T={X,ϕ,{b},{a,b}} be a topological space on the set X.

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: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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Q: 4. Let (X, d) be a metric space and xnx where each x, e X and x e X. Let A be the subset of X which…

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Q: Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set…

A: Given that A is a subset of the topological space X and p is a point in X. Consider a sequence…

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: Let X and Y be two discrete spaces, then * X is never homeomorphic to Y X is homomorphic to Y if and…

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Q: Let (X, d) be a complete metric space and let F ⊆ X be a closed set. Show that F is complete.

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Q: 3. Let K,and K, be compact subsets of a metric space X. Show K, U K2 is compact

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Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

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: If x is a limit point of a subset A of a metric space X, then sh

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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: Let X = R, and d be the discrete metric, and A = [3, 6) Show that if A is compact or not

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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 x be an interior point of a subset S of a metric space (X,d). Show that x must also be a limit…

A: Interior Point: Let A be a subset of metric space X, d. A point x∈A is called an interior point of A…

Q: 6. Let E CX where X is a metric space. Show the limit points of E are the same as the limit points…

A: Limit Point: Let be a metric space. A point is a limit point of a set if for each there exists an…

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: (d) Every continuous map on any metric space is an open map.

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Q: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

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Q: a) Consider the set X = {1,2,3} with the topology t= (0,X, (1). (2, 3}. (1, 2, 3} }. Show that (X,r)…

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Q: Let (X7) be a topological Space and Show that A is open and closed m (xx) iff %3D

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Q: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

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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: 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: Let X be ametric space. If A and B are each dense open subsets of X then AOB is dense open subset of…

A: First we have shown the intersection is open. Then we use the property that if a set intersect every…

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 X be a discrete spaces then * X is never homeomorphic to R O X is homeomorphic to R if and only…

A: Fourth option is correct.