Let Y be an infinite subset of a compact set XC R. Prove Y' 0.

Q: (c) For the set A = {[0], [6]} ≤ Z12 determine the set U = h(A). Then for the set UC Z18 determine…

A: In group theory, sets like Z12​ and Z18​ are examples of cyclic groups under addition modulo 12 and…

Q: 7. (10 points) Let f: A B be onto. Prove that S((C)) = C for all sets CC B.

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Q: 2/ If (xit) is top spaces.t x+y there are two disjoint Pen sets containing them Prove {y? y ex…

A: We use the following theorem  Arbitrary union of an Open set is open.

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

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Q: Suppose A and B are bounded and nonempty subsets of real numbers and 1 E R. In this case: sup(A U B)…

A: Let A,B not empty, bounded subsets of R .Prove thatsup(A∪B)=max{sup(A),sup(B)}. Let: ⇒x∈A∪B⇒x∈A or…

Q: Let u = (2, 3, 1), v = (1, 3, 0), and w = (2, -3, 3). Since (1/2)u -(2/3)v - (1/6)w = (0, 0, 0), can…

A: A set of vectors u1, u2, ……, un is said to be linearly independent if c1u1+c2u2+……+cnun=0 implies…

Q: X, Y and Z be subsets of the set . Verify the following propert XU(YUZ) = (XUY)UZ, XN(YnZ) = (XnY)…

A: X, Y, Z be subset of a universal set Ω to verify…

Q: Let A and B be sets, and let f : A → B be a function. Let S ⊆ B be a subset. a) Prove that f(f −1…

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Q: Show that R is not compact.

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

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Q: 3) Let X be a set with > 3 elements. Show that there exist bijective functions f :X → X and g : X +…

A: Let x=1,2,3 then, f=123132g=123321 `

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: A4For any two finite sets A and B, define f(A,B) to be the number of elements that are contained…

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Q: Prove that interval [0,1] is uncountable.

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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: Prove disprove that (R.Tco-finite) is Te-space or

A: To Prove - Prove or Disprove that ℝ, τco-finite  is T2 - space.       Definition used - T2 - space…

Q: Is the set θ={((x,y),(3y,2x,x+y)) : x,y∈R} a function? If so, what is its domain and range? What can…

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Q: Prove that if XY, then X UZ YUZ for all sets X, Y, and Z.

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Q: Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)}…

A: Given that A is non empty set which is bounded below by 3. Given that B=x∈R|∃a∈A,b=1a The objective…

Q: (x,7) is a topological space, IcCo,1] and {A;liez be Fami ly of subsets of x Uier Ai = JieI A: is…

A: Note that,  Ui∈I  Ai  C— Ui∈I         Ai   (: Ai CAi  for all i∈I) By the condition  Ui∈i Ai is a…