Determine (with argument) whether the following subsets of R are sub- spaces: 1. {(a, b, c) E R³ | a · b+c>0}. 2. {(a, b, c) E R³ | a + b+2 ·c= 0}. 3. {(a, b, c) € R³ | a +2 · b+c= 1}.

Q: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

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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: 4:: what the Null Space of is T;: R3 R³( T,x= xx a) R, R (T = X.Q

A: As per guidelines, we will solve first three sub-question: We have to find the null space of  1.…

Q: 5. If G1,G2 be two given closed sets in a metric space (X, d) prove that: G1 U G2 is closed.

A: We have to prove given result:

Q: 1.* Let U C R? be an open and bounded set and u e C²(U) n C(U) satisfy -1 Ди(х) : in U. 1 + |x|

A: Since I have solved similar ones, Let us show that u attains its minimum on the boundary

Q: 3. Let (X, d) be a metric space and x1, x2, ... , Xn, ... a sequenc

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Q: Let A and B be subsets of topological spaces X and Y, respectively. Prove that CI(A × B) Cl(A) ×…

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Q: Let (N, d) be the usual metric space, then the set A= {1,3} is O closed only O neither open nor…

A: ( N,d) be a usual metric space of natural number.  A={1,3} be a finite subset of N. Since A is…

Q: Let CI(A) means the closure of A.. If A = [0, 1), Bd(A) = {5}, then Cl (A)

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Q: Given the relation R = {(u,t),(v,t),(w,t),(w,u),(w,v)} on a set A = {t,u,v,w}, what is/are the…

A: topic - relation

Q: 18 What is the size of the space stretched by the set {x, x+x², x³, 1 , x²} ?    a) 2nd  B) 34…

A: Solve the following problem

Q: - Let (X, t) be a topological space and A, BCX, then 1- A° 2 A- b(A) 2- A° # A- b(A) 3- A° = A –…

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Q: * Let R be the usual space." In R, if A = [-5,0) U N, then Bd(A) = N O [-5,0] O [-5,0) O [-5,0] UN O…

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Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: 3.5 Show that the metric space (R, d), where d(r, y) = |x – y|, is separable.

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

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

A: Let (M,d) be a metric space and let S⊆M. A point a∈S is said to be an Interior Point of S if there…

Q: Let (X,t) be a topological space, A, B C X. Then A' UB' = (A U B)', A'n B' c (A N B)' Select one: O…

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Q: 1. (a) Define what a linear subset of a projective space is.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: State true or false with brief justification: L1[a,b] is a reflexive space

A: We use the Theorem: Let X be a separable normed space. If the dual space X* of X is non-separable,…

Q: 5. Let (X,T) be a discrete space, then Va E X, the set {a} is а. ореn b. closed c. clopen d. is…

A: Let (X, T) be a discrete space.

Q: Which of the following statements is not true? (i) In any metric space (X,d) every closed ball is a…

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Q: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

A: Given: A subset (a, b)⊆ℝ is T2-space in ℝ, Tu. To determine: Whether given statement is true or…

Q: The set X is an open set in the ambient metric space . O True O False

A: Solution

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…

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

A: Given: τ=B⊆X : ∀x0∈B, ∃rx0&gt;0, such that Brx0x0⊂B is a topology in X. Open ball Br(x)=y∈X : d(x,…

Q: let U and y be bounded sets. if u er i tre M+ (V) ≤M₂ (V) and M^ (U) SM³ (V) 1

A: Given :  U and V are bounded sets with  U⊆ V. To show :  m*(U) ≤ m*V  and  m*U ≤ m*V,  where m*U and…

Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

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Q: suppose X= R¯ is two - dimensional a Space or the same page. X = (X 1,X2) %3D %3D 2 doo (X,y); = max…

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Q: Let (X, M, u) be a measure space wE

A: Given, (X, Μ, μ) be a measure space where μ is a probability measure. Assume thatf, g : X→ℝ are…

Q: Which of the following are compact spaces? (a) S" with a finite number of points removed. (b) The…

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Q: Is the following statement True or False? Justify each answer. If S is a closed bounded subset of a…

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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: For sets A and B, define A+B = {a + b: a € A, b E B}. (a) Prove that if A and B are open, then A+ B…

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Q: . Show that every map f : A → B of sets decomposes as hog where g : A → C is surjective and h : C →…

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Q: What is the size of the space stretched by the set {x, x+x², x³, 1, x²}? a) 1 B) 4 C) 0 D) 34…

A: size of the space streatched   by the set of n element is 2n+2

Q: 5. Let (X,T) be a discrete space, then Va E X, the set {a} is а. орen b. closed c. clopen d. is…

A: Our objective is to choose the correct choice

Q: Amongst the various possible products of four topological spaces A, B, C and D, show that (A × (B ×…

A: We shall solve this in next step

Q: Assume X is Hausdorff. If K C X is compact and x ¢ K, show that there exist disjoint open sets U and…

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Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

Q: For any metric space (X, d) and x EX, the singleton set (x) is closed. Select one: OTrue O False

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Q: Prove that if X,d is a metric space and G ⊆ H ⊆ X, then G-bar ⊆ H-bar.

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Q: Prove that the open ball B(p, 8) = {x : ||x – p|| < 8} is a convex set. [Hint: Use the Triangle…

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Q: Which of the following sets are compact? A={(x,y)| x2+y2 ≤ 1} B={(x,y)| 0 < x2+y2 ≤ 1} C={(x,y)| 0…

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Q: (c) If X={1, 2, 3}. T= {4, X, {1}, {1,2}, {3}, {1,3}}. JUSTIFY whether the topological space (X, t)…

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Q: Find isomorphic spaces O R and R² O P3 and M2x1 O P3 and M2×3 O P3 and R2 O R' and M2×2

A: We have to choose whether the given spaces are isomorphic or not.

Q: Determine whether the following sets form sub- spaces of R³: (a) {(x1,x2, X3)" | x1 +x3 = 1} (b)…

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Q: Every T2 (Hausdor ff) space is T1 space. True O False O

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Q: Let R be the set of real numbers and d: R x R → R be defined as follows. 5|x-y| d(x, y) = 3+2|x-yl…

A: d(x, y) is metric space.

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

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