C.1 Let r > 0 and define the closed ball by B,(x) = {y € X : d(x,y)

Q: The graph of f(x,y)=y2-x2 on XYZ space has a saddle point (a,b) if... Choices: A) D(a,b)=0 B)…

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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: Assume that is a bounded open set in R2 and that g: dR continuous, with the following ball boundary…

A: Given: The given details are, Ω is a bounded open set in ℝ2 and that g : ∂Ω→ℝ continuous with the…

Q: 6. The distance between a point x E R and a nonempty set A CR is defined by d(x, A) = inf {|x – y| :…

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Q: Let P denote the path that travels along the graph of y = x2 from (0,0) to (1,1). Let f(x,y) = √1 +…

A: Let P denote the path that travels along with the graph of y=x2 from (0, 0) to (1, 1). Let f(x,…

Q: 69. Let F(x, y) = (x,0). Prove that if C is any path from (a, b) to (c, d), then F. dr (o2

A: Given: F(x, y) = <x, 0>. C is any path from (a, b) to (c, d).

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 A = {0, 1} with the discrete topology and let f: A R be defined by f (0) = -1, f(1) = 1.

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Q: Let z=x+iy ∈ C where x∈R\{1}, y∈R, and |z|=1. If w = (z+1)/(z-1), then show that Im(w) = y/(x-1).

A: Use the following formulae, to obtain the required result. Real part of a+ib=aImaginary part of…

Q: Let E = [-2, 3) U {−3, π}. Determine the following max D inf D interior of D boundary of D closure…

A: Solution:-

Q: c) For x e ll n2, define || x ]| = || x || 1+ || x ||2: Show that (11 n2, ||· || is a Banach space.

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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…

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Q: Let (X, Tx) represent a topological space and {a} a singleton set, which is an open subset of X, {a}…

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Q: Let X=ℝ2 and define d,:ℝ2×ℝ2→ℝ d2((x1 ,y1),(x2,y)2)) = max{|x1 - x2|,|y1 - y2|}. Draw the…

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Q: In Exercises 1-7, (X,M) is a measurable space. 1. Let f : X R and Y = f-1(R). Then f is measurable…

A: Given: Let's consider X,M is a measurable space.  Here f:X→ℝ¯ and Y=f-1ℝ

Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: 6. Suppose M and N are C manifolds, U an open set of M, and F: U N is C. Show that there exists a…

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Q: 1. Find the absolute maximum and minimum values of the function 2u2) on the set D where D is the…

A: We have been given the function f(x,y)=e-x2-y2(x2+2y2)      (1) on a set D x2+y2≤4 where D is the…

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: 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: Exercise 6.41. Let E be a nonempty set in Rn. Then the distance of a point x € R¹ to E is defined by…

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Q: A player has two continuous choice variables: x € [0, 1], and y E [0, 1] Illustrate the player's…

A: The solution is given below in the next step:

Q: Q/ show thet f(2) = 2+ Conformal maps 121= 1 ? on

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Q: Let R have its usual topology and let X = {a,b, c}. Define f : R –→ X by а, х

A: Given, ℝ have its usual topology and let X=a, b, c. Define f: ℝ→X   by…

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: 4:x→X (when X is a normed Space) is continous at xo € X iff for any Sequeue ie AS n s

A: Let f:x→X (when X is a normed space) is continuous at x0 ∈X iff for any sequence  xn ∈X, xn→х0  ⇒…

Q: (b) If X is a normed line- ball M= {xeX:x≤1}

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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: 2.5 For each of the following properties, give an example consisting of two subsets X, Y CR', both…

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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: 9. Let be a bounded open subset of C, and 4: Prove that if there exists a point zo E such that 2 a…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: 2) Let X= {a, b, c, d} J= {X,Ø, {a}, {b}, {a, b}}. Compute the closure, interior and boundary of A…

A: Here given set X=a, b, c, d and the topology on set X is given as I=X, ϕ, a, b, a, b

Q: Question # 01. Consder z= {acbos a-b =0; S2 Interpet that eguivale nce relation. ~ 18 is

A: That's easy, just show that this relation is reflexive, symmetric and transitive.

Q: 4. Let (x, d) be a metric space and ACX be nonempty set. Define f:X → IR, fix)= inf dex,a). aE A…

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Q: 8- Let X, Y are metric spaces and f: X → Y .If for any A C X, f(A) C f(A) then f is continuous.

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Q: 2. Let S,T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly…

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Q: Let a and b be nonzero and f(x, y) = (ax + by)e-x Show that if a ± b, then there are five critical…

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Q: 4. Let f : R" → R", x+ Ax + b be an affine function, where A E R"xn,beR". (a) Show that if X CR" is…

A: Let f: ℝn→ℝm be an affined map defined as x↦Ax+b, where A∈ℝm×n and b∈ℝm. Then it follows: fx=Ax+b…

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: Consider Uu=4uzzy u(x,0) = u(x), u₁(x,0) = u₁(x), on the domain -∞<x<∞, 0≤t<∞o. Suppose that both…

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Q: 3.27.2. Let X be a metric space with metric d; let ACX be nonempty. (a) Show that d(x, A) = 0 iff x…

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Q: 6. The distance between a point x ER and a nonempty set A CR is defined by d(x, A) = inf {\x – y| :…

A: In this question, we have to prove that d(x,A)=0 iff x belong in closer(A)

Q: 1. Consider the following three domains in the space: {(x,y,z) E R³:1 ≤|x+y+z<4}, {(x,y,z) € R³ : 2…

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Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

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Q: Let d: R2 x R2 R be defined as follows. d((x1, y1). (x2, y2)) = 3/(x1 - x2)² + (y1 - Y2)². %3D Then…

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Q: 1- Suppose that ,p) is a meric space, an that E CX. Then the interior E is open (m X) and the…

A: The given question is related to metric spaces. The solution is explained below.

Q: IIC sUUset of R". tve R" and let k e R. Prove that S = {x e R":x.v = k} an affine subset of R". %3D

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Q: 3 \et F be anormid space and- |2+j|l <llxll+yllprove that F has a stridtl (3) y convex- orm

A: We know that a normed space is strictly convex if and only if x≠y, ∥x∥=∥y∥=1, we have ∥x+y∥&lt;2 Let…

Q: Prove that if w = f(x,y,z), x=r-s, y=s-t, and z =t-r then ôw ôw ôw + âr' as = 0. ôt

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Q: 3. Determine whether the following sets are open, člosed, both open and closed. Also, decide which…

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