Proposition 1.35. The intersection of any family of closed sets in a metric space X is also closed in X.

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

A:

Q: 5. If G1,G2 be two given closed sets in a metric space (X, d) prove that: G1 U G2 is closed.

A: We have to prove given result:

Q: 3. Let (X, d) be a metric space and x1, x2, ... , Xn, ... a sequenc

A:

Q: Let (X,T) be a toplogical space, A, BC X. Show that АС В Ас в, АсВ. 介

A:

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

A:

Q: For any topological space (X, T) and A C X, the set A is closed. That is, for any set A in a…

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

Q: Theorem 2.8: The closed subsets of R have the following properties: (a) R and Ø are closed sets. (b)…

A: Note : A set is said to be closed if it's complement is open. Theorem used: Arbitrary union of open…

Q: 2. (a) Use axiom (2) in the definition of a topology to show that if {U1,U2, . ,Un} is a finite…

A: It is given in part (a) that (X,t) be any topological space, and the interval (x-A,x+A) is only open…

Q: 6. Show that the span of this set is all of R?. {}}

A: Let S=v1,v2,...,vn be a set of vectors in a vector space V(F) . The vector space V is spanned by S,…

Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: If a < b, then [a,b] CR does not have con- tent 0. In fact, if U1,. . .,Un) is a finite cover of…

A:

Q: Which one of the following statements is true? * R with the Euclidean topology and R with the finite…

A: Answer

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

A:

Q: For all sets A , B and C A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C )

A:

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

A:

Q: 7 1-{XEN: with the usual metric is connected. 2 The set M= {Xe O True False

A: Given a set M=x∈ℕ | x≤72. Clearly, M=1, 2, 3. Suppose that d is a usual metric. Then, let ε=mind1,2,…

Q: If A C BỊ U B2 where B1 and B2 are disjoint open sets and A is compact, show that An B1 is compact.…

A: To Show- If A ⊆ B1 ∪ B2 where B1 and B2 are disjoint open sets and A is compact, show that A ∩ B1 is…

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: Which of the following statements is not true? (i) In any metric space (X,d) every closed ball is a…

A:

Q: Suppose metric space X is not path-wise connected then X is not connected. O True O False

A: Question 9 Answer: as a path connected space of employees connected but converse is not true. So I…

Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

A:

Q: The set X is an open set in the ambient metric space . O True O False

A: Solution

Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: In any metric space, the only open and closed set at any time is the empty set and X only True False…

A:

Q: Proposition 1.36. For any metric space X, the empty set o and the whole set X are closed in X.…

A: For any metric space X, the empty set ϕ and the whole set are closed in X

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.

Q: 2. Show that the set 3 S = 3 2 2 2 2 spans all of R³ or find an element of R' not belonging to…

A:

Q: ( Prove that the set {a E R2: a2 ... Prove that the set {a E R2: a2 not equal to 6} is open.

A: Let the set is A=a1,a2∈ℝ2:a2≠6 The objective is to prove that the set A is open.

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

Q: Proposition 4.4.6. A subset A of a metric space S is closed if and only if A' C A. When we…

A:

Q: 2.6.9) The empty set Ø is a subset of every metric space X. What are the interior, closure, and…

A: The empty set ∅is a subset of every metric space X To determine interior, closure and boundary of…

Q: Proposition 1.46. For each i in some indering set I, let A; be a subset of the metric space X. Then:…

A:

Q: 14. (Axioms of a metric) (M1) to (M4) could be replaced by other axioms (without changing the…

A:

Q: Prove that in any metric space, an ɛ- eighborhood of a point is an open set.

A:

Q: The set {0,3} with the usual metric is connected as union of connected sets {0} and {3} True False

A:

Q: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.

Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

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

A:

Q: Proposition 1.36. For any metric space X, the empty set o and the whole set X are closed in X.

A: 1. Let X be a metric space with metric d. If is a point of X and r is a positive real number, the…

Q: Suppose {p, p,, P3} is an affinely independent set in Rn and q is an arbitrary point in Rn. Show…

A: Here, {p1,p2,p3} is an affinely independent set. So, {p1,p2,p3} are linearly independent.

Q: 4.1 Prove that in any metric space (S, d) every closed ball S,ro) is a closed set. 4.2 Let F be a…

A:

Q: 2.2 Is any discrete space complete? Give an example to substantiate your answer.

A: Yes, any discrete space is complete.

Q: Is any discrete space complete? Give an example to substantiate your answer.

A: A metric space is complete if and only if every Cauchy sequence is convergent.

Q: Exercises: 1) In a metric space (M,d). show that both Ø and M are open sets in M.

A:

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: Question 2: Use set builder notations to prove that: a)X Y = (XUY) – (XnY), b)(Y – X) U (Z – X) = (Y…

A: Part(a) We have  by the definition of symmetric difference : X⊕Y=xx∈X-Y or x∈Y-X Now, we prove that…

Q: We defined the relation between sets by A - B means that there there is a 1-1 correspondence f : A →…

A: The complete solutions is in given below

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

A:

Q: If K is a compact subset of RP and z is a point of RP, then the set K, = {r+y :y € K} is also…

A: Given that K is a compact subset of ℝp and x is a point of ℝp. The objective is to show that…