6. Let (X, d) be a metric space and Y CX be nonempty. Verify that (Y,d) is a metric space. Show that a subset S C Y is open in Y if and only if S = AnY for some set A which is open in X.

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: Show that every metric space is an Hausdorff space.

A:

Q: Prove that if the space X' of a normed linear space X is separable (as a metric space) then X itself…

A:

Q: Every totally  bounded metric space is compact

A: This statement is not always true.

Q: 10. Let (X, 71) be a Lindelöf space and (Y, T2) be a compact space. Then show that the carte- sian…

A:

Q: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

A: T2-space: A topological space X is said to be T2-space for given x, y∈X, there exists open sets U…

Q: every metric space is normed space? give an example to show tha

A: We have to show that every metric  space  may not be a  normed  space.

Q: Ex: Let d is a trivial metric o space ) show that every infir

A: We have given trivial metric space d on infinite set X. We have to show that every infinite subset…

Q: Determine if the following statements are true or false. Briefly explain why. a) A quotient of a…

A:

Q: let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B…

A: Let (X,t)  be  not  connected.We  have  to  show  that  X=A∪B ,where  A and B are  elements  in t…

Q: Every metric space is a hausdorff space .

A:

Q: 5. Let (X, d) be a metric space. i) Show that the open ball B, (ro) = {x € X : d (T, xo) < r} is an…

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

Q: Let

A:

Q: a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b.…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: Every T3 space is regular space. True False

A: Since a regular T1 space is called a T3 space.

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

A:

Q: Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only…

A: Given: X,d is a metric space, x∈X and A⊂ X is a non-empty set. To Prove: dx,A=0 if and only if every…

Q: Exercise. Prove that in a metric space (X, p), a set G C X is open if and only if it is a union of…

A:

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…

A:

Q: Exercise. Prove that every finite subset of a metric space is closed set.

A: We have to Prove that every finite subset of a metric space is closed set.

Q: Prove or Disprove the following: a) The product Space of two Connected spaces must be connected.…

A: As there are two proofs asked, I will solve only the first one. Let M1 and M2 be connected spaces.…

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

A:

Q: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: Let (C[a,b] .d) be the discrete metric space. Then Cla,b] is bounded. * False True

A:

Q: 1. Let M, d be a metric space with A C B C M. Suppose that B is totally bounded. Prove that A is…

A: Let (M, d) be a metric space and let A be a nonempty subset of M. The diameter of A is defined as…

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: 15. Let X be a normed linear space. Show that X is finite-dimensional if and only if every closed…

A:

Q: Recall that RK denotes the real line with the K-topology. Let Y denote the quotient space obtained…

A: Given:-             Note that if U opens in R, and then U-K is opens in R k.             U = union…

Q: Exercise. Prove that in a metric space (X,p), a set G CX is open if and only if it is a union of…

A:

Q: (a) Prove that every nonempty proper subset of a connected space has a nonempty boundary. Is the…

A: As per Bartleby guidelines, for more than one question asked, only first is to be answered. Please…

Q: Let (R, d) be discrete metric space and A = N then A is complete and compact neither complete nor…

A: Given that (ℝ,d) is a discrete matric space and A=ℕ. then     d(x,y)=0 if x=y1  if x≠y A matric…

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

A:

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

A:

Q: show by an example that a nonempty proper subset of S in the metric space (S, d) can be both open…

A:

Q: Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

A: Let A be a closed bounded subset of a metric space (X, d). Is A acompact subset? Explain why.