2. Let (X.d) be a metric space. Then consider each of the following cases: a) Suppose ASX contains finitely many points of X. Show that A must be bounded. b) Suppose that R.S C X such that both R and S have finite diameter. Prove that the set RUS must be bounded.

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: Let (X,d) be a metric space and E ⊆ X. Prove that if E is compact, then E is bounded

A:

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

A:

Q: let and be nonempty metric Space x E X. Ihen A B Sebsets of and let d(x, A), dlA, B) and dlA) are…

A: As per bartleby guidelines for more than three subparts in a question only first three should be…

Q: 3.5 Show that the metric space (R, d), where d(r, y) = |x – y|, is separable.

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

Q: Let p1 and p2 be points in a metric space X. Prove that the set p1 ⋃ p2 is compact in X. Now, let…

A:

Q: Let

A:

Q: B. Given a metric space M with metric d, verify that any ε-ball is an open set. (Please If possible…

A:

Q: A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: Given: H is the set of all points x , y in R2 such that x2+xy +3y2=3. To Show: H is a closed subset…

Q: rsection of any collection of com

A: This is a real analysis question. Let (X,d) be a complete metric space with X not ∅. Suppose the…

Q: E. Let H be the set of all points (x, y) in R² such that x² + 3y² = 12. Show that H is a closed…

A: We have to find H is closed  subset bounded.

Q: Exercise. Suppose that (X, p) is a metric space. a) Prove that any set in X consisting of precisely…

A:

Q: Consider the following topological spaces: Space X: A closed unit interval [0, 1] with the usual…

A: Two topological spaces X and Y are said to be homeomorphic if there exists a continuous bijection f:…

Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

Q: Show that in R with the euclidean metric, any subset of R \Q with more than one element is not…

A:

Q: Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: As per the question we are given the following bivariate function : x2 + xy + 3y2 = 3 And a set H is…

Q: Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: Given That : Suppose H is the set of all points x , y in R2 such that x2+xy +3y2=3. To Show : H is a…

Q: B) Let X = R be a metric space and consider the following subsets of X, %3D 1 1 S = (0,1), S2 %3D…

A:

Q: Theorem 2.25. Suppose that (X, p) is a metric space. (a) The intersection of any collection of…

A:

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

A:

Q: Given a metric space M with metric d, verify that any e-ball is an open set.

A:

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

A:

Q: Q7: Prove that ( IR", ||·|| ) is complete metric space.

A:

Q: Let (R, d) be discrete metric space, then R is not separ able O True O False

A:

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: 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: 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 (X, d) be a metric space. The diameter of a non-empty subset M of X is defined as 8(M)= sup d(x,…

A:

Q: If A is a subset of a me

A: Given is a matric space X and a subset A. Then it is needed to prove that  A=x: each open sphere…

Q: Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: Given that: H:(x,y) ∈ R2 | x2+xy + 3y2 = 3Now as (x2+xy + 3y2 = 3) is a bivariate polynomial, it is…

Q: Let A be a subset of X, and B be a subset of X, where X is a metric space. Show the intersection of…

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