5. Let X be a set and p E X. b) If 72 = {A C X : p ¢ A}U{X}is a topology on X, then determine whether (X, 72) is a normal space or not.

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

A: 3.3 is also known as Hausdorff Property. d(x1, x2) > 0 for every x1 ≠ x2   (metric space…

Q: If τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normal space…

A: τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normalspace or…

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

A:

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

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: 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: If there exists open ball B (a, r) such that B (a, r) NG= Ø then a is not limit point of G True O…

A: TRUE

Q: A metric space (X , d) and a collection of open sets U such that UEU is not open.

A: Let X, d be a metric space. Let U be the collection of open sets in the metric space X, d. That is,…

Q: Let (X, E, µ) be a measure space. Justify through an example that the hypothesis µ(X) < o in…

A:

Q: A ° = A – JA

A:

Q: We define the included point topology by Tp={UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R if

A: In order for a set to be dense, it is required to be equipped in Tp and also as in the set 5 is not…

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

A:

Q: (1) Let X be a set and tEX, let t={GCX/ t€G or G° finite }, show that whether t is a topology on X…

A:

Q: In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and…

A:

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

A:

Q: In a Cofinite topology of X the collection of subsets a) r = {AcX|A ±Ø or A is finite} b) r = {Ac X|…

A: Let X be a non-empty set, then the collection of subsets of X whose complements are finite along…

Q: Let d and e be metrics on a set X such that for each ball Ba center at pE X there exists a ball Be…

A:

Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

A:

Q: 2. Let X = {a, b, c, d, e, f}. Which of the following collections of subsets of X is a topology on…

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Q: * Let R be with the discrete topology. If A = {1, 3,5, 7, .}, then Aº A O N O RO

A:

Q: Show that a set {V1, ..., Vp} in R" is affinely dependent when p > n + 2.

A:

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: We define the included point topology by Tp={ UcR;U=ø or peU}. Let A = [3,5[, then A is dense in R…

A: In order to be dense in R, R has to be equipped in the Tp. Also as in the interval, 5 is not…

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

A:

Q: Let (X,d) be any metric space, then the only open sets and closed sets are and X O True O False

A: Given statement is false because if we take descrete metric space

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 C(X,Y) have the compact-open topology. Show that if Y is Hausdorff, then C(X, Y) is Hausdorff

A:

Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

Q: LBe M, N subsets of space W. Then span (MUN) = span (M) + span (N)“ %3D

A:

Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

Q: if

A: R is not connected if T is the finite closed topology.

Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

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: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

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

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: Show that the closed ball Y in a metric space (x,d) is a closed set .Also show that if (x,d) is…

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: consider R with the usual topology then a subset A={1,0.5,..} is: empty or closed or open or none…

A: Given R with the usual topology and given a subset  A=1, 12, 13,.....

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

A:

Q: 3) let X be discrete topology and Ac X, find A ?

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

A:

Q: f A∩B≠∅ and A∩Bc≠∅ then show that A∩X\B̅≠∅ and A∩B̊≠∅

A:

Q: 4. Let X= {a,b,c} with the topology r ={X,ø,{a},{b},{c},}{a,b},{a,c},{b,c}}. Verify whether X is…

A: Given data X=a,b,c with the topology,  τ=X,ϕ,a,b,c,a,b,a,c,b,c

Q: 1/ For any. topological space (X, T), if T has only three elements then (X, T) is connected.

A: Given: Topological space X, T and T has 3 elements  To prove that X, T is connected when T has 3…

Q: Let X = {a, b, c, d, e} and let T ={X, Ø, {a}, {c,d}, {a,c,d), {b,c,d,e}), therm %3D O (X,T) is…

A:

Q: Let M be a closed subset in (X,d). Show that b(M) is a subset of b(b(M)) where b(M) is the boundary…

A: Given: M is a closed subset of X,d. To prove: bM is a subset of bbM, where bM is the boundary of M.