Let T_us be the usual topology on R and T_ll be the lower limit topology generated by the unions of {]a,b]/ a,beR;a

Q: Let X (a, b, 1, 2, 3) and T = {X, 0, {a), (1,2), {a,1,2), {b,1,2,3}). Let A = {a, b, 1), then A (the…

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Q: Give an example if f : (0,1) -> R that is bounded, continuous but not uniformly continuous on (0,1).

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Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

Q: Let R = - b ,be Z}and let ø : R→Z be a mapping defined by : b = a-b. a a

A: We will find out the required value.

Q: The linear map A:R^4->R2[t] is defined by A(x,y,z,u)->(x-z)t. Then the dimension of the kernel…

A: We need to find the dimension of the kernel of the given map, A:R4→R2​[t] defined by…

Q: Exercise 8. Any map f: X Y between topological spaces which is continuous at x = X is sequentially…

A: Continuity and sequential continuity are defined below

Q: Suppose Y=[ 0 ,2 ) . In this case A=[1,2 ) is not closed in X; while closed in Y. In addition, G=[0…

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Q: Let f : A → B be a function. Show that the set P = {f ^b | b ∈ B} is a partition of A.

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Q: Consider a linear map T : R″ → R" with the property that null(T) = range(T). (a) For n = = 2,…

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Q: Let f: X->Y and g:Y->X be maps between sets. Suppose that g ∘ f is equal to the identity map id_X:…

A: Given f:X→Y and g:Y→X be the maps between the sets.  Suppose that g∘f=idx. Then We need to prove or…

Q: (C0]) }

A: The statement is false. Consider f:[0,1]→ℝ defined as f(x)=tan-1πx2   x∈[0,1)0              x=1

Q: Find a condition on a, b, c so that (a, b, c)  ∈ R^3, belongs to the space spanned by →u = (2, 1,…

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Q: f If Xo's Compact, x=y Continuous bijective L Topelagied space Then f is homeomorphism

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Q: 4) Let T be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (R, 71) →…

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Q: Prove that the Peano Curve is continuous and surjective mapping [0,1] onto the unit square

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Q: Let T us be the usual topology on R and Tl be the lower limit topology generated by the unions of…

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Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

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Q: Let ƒ: R² → R² be an isomorphism where ([-])-[²] and f([:]) = [] = Find f ¹ƒ( [7¹]) X 0 2 2 0

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Q: Q. Let X = {1,2, 3, 4},T = {ø, X, {4},{1,2}, {1,2, 4}} and Y = {r, s, p, q}, F = {ó,Y, {p}, {r, s},…

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Q: let ( X Td)be Toplog9 me trics spaces the n EVery singteon set is closed set

A: Given that X,Td is a Topology Metric spaces.

Q: Exercise 8. Show that a function F: R" → R" preserves the norm || · || if and only if it preserves…

A: We have the function F:ℝn→ℝn that preserves the norm · This implies that this function satisfies all…

Q: Q₂/(a) Let X=(1, 4, 5,7, 8, 9) and o= {(1,4),(4,7, 8}}, show that whether o is: (1) a subbase for a…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: الموضوع show that if foret A frix by and ; YーZ is a continuous map then gFo relA gf,:X b Z

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Q: With the help of exomples Show thaat on finite dimensional are equivalent. any two norms Linear…

A: Given: Any two norms on finite dimensional linear space are equivalent.

Q: Let X = R. Define an equivalence relation ~ on X by x ~ y if %3D and only if x – y is an integer.…

A: Let X, 𝛕X be a topological space, and let ~ be an equivalence relation on X. The quotient set,…

Q: Let: f : R → R with f(x) = 2x ^2 + 1. Prove or disprove that f is injective (one-to-one).

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Q: Let T_us be the usual topology on R and T_|| be the lower limit topology generated by the unions of…

A: A homeomorphism is a bijective continuous map whose inverse is also continuous

Q: Lot X= [a,b] with the standard topology and let Y= [9₁6) with the Cofinito topology. Why is the…

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Q: Consider the topology T on X= (a, b, c, d, e), (X, Ø, (a), (e,d}, {a, e, d), (b, c, d, e}} Show that…

A: It is given that,  X = {a, b, c, d, e} and  T = X, ∅, a, c, d, a, c, d, b, c, d, e be a topology on…

Q: Let T_us be the usual topology on R and T Il be the lower limit topology generated by the unions of…

A: Homeomorphism is a bijective continuous map whose inverse is also continuous

Q: B\ If X is Pre-Hilbert space then (x,ŻB,Y, ) = ΣP, (x,y, (x, j=1 j=1

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Q: Prove that the Joukowski map 1 p(2) : (z + maps ONTO T - (-1, 1)

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Q: Show that is not compact for the Standard topology but is compact for the cofinite topology.

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Q: f: R →R by f(x)= x^2 and V is a subset of the codomain. Find a set V such that V has only one…

A: We are given that f : R → R by f(x) = x2 and V is a subset of the codomain. We need to find the set…

Q: Let T us be the usual topology on R and T_Il be the lower limit topology generated by the unions of…

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Q: Define a linear operator on R^2 by performing the following operations in order: i) a shear in the…

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Q: 8- Char (R) = a) Char (R) b) Char (Q) c) Char (Z) d) None O a) O b) O c) O d)

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