Let M, be set of all rare subsets of a metric space X. Prove that countable union of M, is rare.

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: 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 S is compact and D is in S is a closed set, then D is compact. Use the definition of…

A: Let S is compact and D be a closed subset of S. We are to prove that the set D is compact in S. Now…

Q: Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

A:

Q: 3- In a complete metric space (X, d), If A C X such that, A = A then A is complete as a sub space of…

A:

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

A:

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: A subset I of a metric space R with the usual metric is compact if and if only it is an interval…

A:

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

A:

Q: Prove if {Galpha}alpha e A is a collection of closed sets in a metric space, then…

A: Recall that arbitrary union of open sets in X is open in X.

Q: For a normed space X and a closed convex subset A of X. Prove that (º )nÎ= Â. A

A: Let, be a normed space and be the closed convex subset of As, we know that norm on is a function…

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

A:

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: Let X be an infinite set. (a) Show that is a topology on X, called the finite co со

A:

Q: 3. Prove that if G is an open set dense in the metric space (S, d) then SG is nowhere dense in S.

A:

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: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology.   We claim that Z is NOT…

Q: Let A1,·.., Am be compact subsets of R. Prove that Aı × · … × Am is a compact subset of Rm.

A: Given, A1,A2, ... , Am are compact subsets of R. We prove A1x A2 x ... x Am is a compact subsets of…

Q: Let

A:

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: 6 Prove that a closed set in the metric space (S, d) cither is nowhere dense in S or clse contains…

A:

Q: 1) In a metric space (M, d), show that both 0 and M are open sets in M. 2) In a metric space (M, d),…

A:

Q: Let (Ka)aeA be a collection of compact subsets of a metric space, X. Show naeAKa is closed in X.

A: A metric space X is compact if every open cover of X has a finite subcover.

Q: For any infinite set X, the co-countable topology on X is defined to consist of all U in X so that…

A:

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

A:

Q: Prove that for any infinite index set J, the uniform topology on R° is strictly finer than the…

A:

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: 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: 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 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: 4. Let {Ka}aeA be a collection of compact subsets of a metric space, Show NaEA K, is closed

A:

Q: 1) In a metric space (M, d), show that both 0 and M are open sets in M. 2) In a metric space (M, d),…

A: Open sets :- Let (M, d) be a metric space then a subset A of M is open if and only if each point of…

Q: in the discrete metric space: 1. All collections are clo

A: Let (X, d) be a metric space. Consider a closed ball Brx0 with both center x0 ∈ X and radius r >…

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

A:

Q: Let (X.J) be a topolgical space and {C1, C2 Cn} be a finite collection of compact subsets of X.…

A: Given Let (X,I) be a topological space and C1,C2,......Cn be a finite collection of compact subset…

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

A:

Q: Let K be a compact subset of a metric space X, and assume that -R is continuous. Prove that f…

A:

Q: (a) Prove that every closed subset S of a compact metric space (M,d) is compact in M.

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

Q: Let X be a set with more than one element. (X,7) is a topological space such that Vxe X, {x} €T then…

A: Given the set X contains more than one element.

Q: Let ||.|| and ||..|| be two norms defined on space X.Show that ||.|| and ||..|| are equivalent if…

A:

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

A:

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

A: Given that X,τ is a topological space. Let's prove this theorem by contradiction. So  X,τ is a…

Q: 2- In a metric space (X, d), A is the intersection of all closed subsets of X which contains A.

A:

Q: Let (X, d) be a metric space and let A be a non-empty subset of X. Prove that A is open if and only…

A:

Q: Let A be dense subset of (X, t)

A: Given: Let A be dense subset of x,Τ and let B be non empty open subset of X. To find: Fill in the…

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

A:

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: Let x1 and x2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A: