Let the non-empty set S {x-y3:x,y E R}, show that S is closed'under multiplication then find the identity element.

Q: 3. Let r>0 and define the closed ball by B,(z) = {y € X : d(x,y) Sr} Show that B,(r) is closed set…

A: Given X be a metric space with metric d. Now define a topology τ on X as follows: τ=B⊆X:…

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y.

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Q: Consider the set P = {f€C([0, 1], R) | ƒ is nondecreasing} Determine if (P,d2) is separable. Here,…

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Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. *

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Q: f(x,y) = (In(y + x²)) x² + y² D = {(x,y) ERIy2x²-1 and y ≤4-e-*) (b) Observe the set Is D…

A: (b) The given set is D=x,y∈ℝ : y≥x2-1 and y≤4-e-x To find: The given set is bounded or closed or…

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)…

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Q: Suppose M and N are C manifolds, U an is C. Show that there exists a neighborhoo that F can be…

A: Let us start solving the problem step by step

Q: Let (X,T) be a discrete space, then Va e X, the set {a} is а. орen b. closed c. clopen d. is neither…

A: Given problem is :

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

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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: E be a Euclidean space of dimension (E(1, f)) = dim(Ker (f- id)), then s minimal.

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Q: For every x, y E R, let p(x, y) = |x² – y²\. Is p a metric on R?

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

Q: Let R = [0, 1] × [0, 1] . Est each with at least six rectan IR 6(x + y) dA 2

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

Q: Let X be an uncountable set and let oo EX be fixed. Let TX = {U E P(X); 0$U or X- U is finite} Let Y…

A: Given: Let us consider the given X to be unaccountable and let ∞∈X be fixed.

Q: 3. Let r>0 and define the closed ball by B,(1) = {y €X : d(r,y) Sr} Show that B,(r) is elosed set…

A: Given that X be a metric space with metric d. The given topology on X is defined as:…

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

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Q: (f) The set B = {(x, y) E R² : x² + y² < 1} as a subset of R? with indiscrete metric is…

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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: Consider the sets X = {(x, y) E R²: x² + y² ≤ 1) and Y = ((x, y) = R²: x ≥ 0]. Sketch on R²the sets…

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Q: Let, T:R' → R';T(x,y,z) = (x + y+ z,2y+z,2y+3z) 2. Find the dimension of KerT and ImT

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Q: Let (X,d) be a metric space. Prove that the function d' :X x X →R given by d (x, y)= min {1,d(x, y)}…

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

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

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

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Q: LetX={(x, y)∈R×R:xy= 1}.Suppose thatR×Rhas thestandard topology, andXhas the relative topology.…

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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…

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Q: Let (X, T1) and (Y, T2) be topological spaces. Show that the set {U × Y ∣ U ∈ T1} ∪ {X × V ∣ V ∈…

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Q: 3. Find the closure and interior of the set A = {(x, y): xy = 1} as a subset of R? (equipped with…

A: Closure and interior

Q: an open ball in the set X, with its radius & satisfying -d(x, a), Vx € X at the open ball Be (x)…

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Q: Let (X, d) be a metric space and B = S(xo,r) be the open ball with centre at xo E X and radius r> 0.…

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Q: Let A = Z. Let R be defined on A x A where a Ry if 4 (x- y). Prove that R is transitive.

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Q: (f) The set B = {(z, y) E R² : ² + y < 1} ; is disconnected. as a subset of R? with indiscrete…

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Q: Let X be an inner product space over C and x, y = X are such that || x + y || = || x-y ||, then show…

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Q: Let T be the co-finite topology on R and d0(x,y)=|x-y|. Prove that the function f:(R,T) to (R,d0)…

A: Given that T be the co-finite topology on ℝ and d0x,y=x-y To prove that the function f:ℝ,T→ℝ,d0…

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: 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: Let 1={Ø}U{UER:NC U}, -n+3 ). A = :n EN and B = [0, ∞). Then n+1 in the topological space (X, 1) O…

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Q: For every ,x, y E R, let p(x, y) =| x² – y² |. Is p a metric -

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Q: 4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show that E is closed.

A: Let (X,d) be a metric space, with x∈X , ε&gt;0, and E={y∈X:d(x,y)≤ε} To show: E is closed. That is…

Q: Define on R, the metric d defined by d(x, y) = |1+1리 1+ |y| Then (R, d) is not complete. True False

A: Given

Q: Let, T: R' → R';T(x, y,z) = (x+ y+ z,2y + z,2y +3z) Find the dimension of KerT and ImT

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

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Q: Let R have its usual topology and let X = {a, b, c}. Define f : R → X by x 0 а, f(x) = с, Is X with…

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Q: Consider the space L'[0,1]. Let p(f) :

A: Given, pf=∫1234f212, f∈L20,1. Standard L2-norm is f2=∫01f212

Q: Let X = R, and let T consist of 0, R, and all intervals of the form (a, b) where a, b e R and a < b.…

A: A topological space is an ordered pair {\displaystyle (X,\tau ),} where {\displaystyle X} is a set…

Q: Consider the set R. Prove: (i) VxER, x 0 = 0;

A: To prove: For any x∈ℝ, x·0=0.

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

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Show f is…

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Q: A° U B° c (A U B)

A: