Let X be a topological space and A be a subset of X. Prove that (A°)° = Ac and (A) = (A°)°.

Q: Let X be a normed space and aɛ X. Show that there exists an fe X' such that ||a|| = Sup { \f(a)| :…

A: Given that,                X is a normed space and a ∈ X . We have to show that there exists an f ∈…

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: Let (X, d) be a metric space and let A ⊆ X be a compact subset. show that, If B ⊆ X and d(A, B) =…

A:

Q: For any subset A of a topological space X, we define the boundary of A to be: JA = AnX\A. a) Show…

A:

Q: Let V be an inner product space, and suppose that x and y are orthogonal elements of V. Prove that…

A: Given:

Q: A point y in space either belongs to a given closed c ough y so that whole of the x lies in one open…

A:

Q: Let V be an inner product space, and suppose that T: V→V is linear and that ||T(x)|| = ||x|| for all…

A: Given:

Q: Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B)…

A:

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…

A: TRUE

Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

A:

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: Let (X,7) be a topological space and ACX. Show that (n) A is open if and only if Int(A) = A. (b) A…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: The space X = {0,1} with topology T = {0, {1}, {0, 1}}) is To but not T Select one: O True O False

A: Discrete spaces are always T0 , So X is T0 .  A space X is a T1 space if and only if  for each x∈X,…

Q: Let X be a disconnected space, that is X = UUV, where U and V are nonempty open subsets of X and…

A:

Q: For any subset A of a topological space X, we define the boundary of A to be aA = AnX\A. a) Show…

A: (a)

Q: Prove the following. Let J be any set, and let {Xa : a E J} be an indexed family of connected…

A:

Q: Theorem 2.28. Let A be a subset of a topological space X. Then Int(A), Bd(A), and Int(X – A) are…

A: Given that A is a subset of the topological space X. We have to prove that IntA, BdA and IntX-A are…

Q: Let (X, T) be a topological space and A, B be a subsets of X then: اختر احدى الدجابات int(A - B)=…

A:

Q: The collection {a} × (b, c) C R² | a, b, c E R} of vertical intervals in the plane is a basis for a…

A:

Q: Let ( Xr d be ame tric space,show that exist atopology on X induce b

A: Let (X, d) be a Metric Space We know that in a Metric Space  (1) arbitrary union of open sets is…

Q: 2. Find the complement CC of the set C with respect to the space C if (a) C = {x :0 < x < 1}, C = {x…

A:

Q: Prove the following. Let J be any set, and let {Xa : a E J} be an indexed family of connected…

A: It is given that Xα:α∈J is an indexed family of connected topological space that is each component…

Q: Let (X, T ) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f ∈ C(X, M);…

A: Given as X,T be a topological space,M,d be a complete metric space and BCX,M=f∈CX,M;fX is bounded…

Q: If M and N are metric spaces, the Cartesian product M x N is again a netric space with the distance…

A: We have given that , M and N are metric spaces. We have to show that , M × N is again a metric space…

Q: Show that R is a bianch space.

A:

Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

A:

Q: Let v e R" and letk e R. Prove that S = {x € R" :x•v = k} is an affine subset of R". %3D

A:

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

A:

Q: Let X be a finite dimensional norm space. Then prove that M = {x € X |||x|| <1} is compact.

A:

Q: let C be a compact subset in a Hausdorff topological space (X,T) and b∈X a point. If b∉C then prove…

A:

Q: ) Let X be a finite dimensional norm space. Then prove that M = {x EX |||x || <1} is %3D compact.

A: Sufficient to prove that M is closed and bounded

Q: Let X be a normed space and A € BL(X) be of finite rank. Then σe (A) = σ₁(A) = o(A).

A: 1

Q: Let X be a topological space. Construct a topological space Px and a continuous surjection 0: X→ Px…

A:

Q: E be a subset of a normed space X and Y = Span E and a € X. Show that Let a € Ỹ if and only if f(a)…

A: To find: a∈Y¯, and f(a) = 0 whenever f∈Xι and f is 0 everywhere on E.

Q: Let A be a non-empty subset of the metric space (X,d). show that a. d(x, A) – d(y, A)| < d(x, y)For…

A:

Q: Let {Xa | a E A} be a collection of Hausdorff topological spaces. Prove that IαEA Xa is Hausdorff…

A:

Q: Prove that Let X be a metric space such that X E1 U E2. If E1, E2 are complete, then X is complete.

A:

Q: Let A be a sSubset of a metric space A. Given r€A, define the distar from x to A to be dist(z, A) =…

A:

Q: If the unit sphere {xe X : x = 1} in a normed space X is complete, prove that x is complete.

A: Given that X is a normed space. Let S1x=x∈X| x=1 be unit sphere of the normed space X. Given that…

Q: Let X = {a.b.c.d.e) Find the topology r on X generated by f-{{a.b.c).(b.c.d).fc.d}}

A: X={a,b,c,d,e}

Q: Let E be a Euclidean space with inner ratic form. For all u, v E E, we have the C y(u, v)² ≤…

A:

Q: If X is a metric space and Y C X, then Y is a metric space.

A:

Q: Let (X, d) be a compact metric space and T: XX be a mapping that satisfies d(Tr,Ty)<d(r,y) for all…

A: What is Metric Space: Metric space is an abstract set in mathematics, particularly topology, with a…

Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

A:

Q: Let E and F be compact sets in a metric space (X,d). Show that E U F is compact using the definition…

A:

Q: (b) Describe graphically the set A={(x,y):d[(x,y), (0,0)]<3}for the following metric space on R²…

A:

Q: Let X is a normed linear space and the ball M= {xe X:|| x ||≤1} is compact, then X is finite…

A:

Q: A° U B° c (A U B)

A:

Q: Prove that any smooth manifold (M, A) has a countable smooth atlas V (so that D(V) = A)

A: