A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

Q: Let X be any infinite set with the co-finite topology. Identify which of the following separation…

A: Let X be any infinite set with co-finite topology. Definition:  Let X be any infinite set. The…

Q: Consider R×R to be a universal set with subsets A, B, and C defined as follows. = {(x,y)x* + y° s1}…

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Q: Exercise 4.4.7. Let A be a set. Let Ø : P(A) → P(A) be defined by ø(X) = A − X for is bijective. all…

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Q: Let X be an infinite set with the countable closed topology T={S subset of X; X-S is countable).…

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Q: 1. Let Ω = (0, ∞). Let A = (3, 8] and B = (5, 10]. Find the smallest collection of subsets of Ω that…

A: Definition: Let Ω be a non-empty set. Let Σ be a collection of subsets of Ω. The collection Σ is…

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Q: Consider the set T = {..., −34, −17, 0, 17, 34, ... } with the usual definition of less than (a0).…

A: Completeness property of real numbers:  Every non-empty set of real numbers that has an upper bound…

Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: e) Does the equality |A U B| = |A| + |B| hold for every set A and B? f) Is it true that for any…

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Q: (1) Let X be a set and tEX, let t={GCX/ t€G or G° finite }, show that whether t is a topology on X…

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Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

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Q: Give an example of a set X and topologies T1 and T2 on X such that T1 union T2 is not a topology on…

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Q: Let X be an infinite set. (a) Show that is a topology on X, called the finite co со

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Q: The continuous image of a connected set is connected. That is, if X is connected and f: X→ Y is…

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Q: Let X be a non empty set and p be a fixed point in X. Define T={X, ACX such that p€A}. Then any…

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Q: (b) Let F be a non-empty subset of R. Prove that if OF C F, then F is closed.

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Q: Every bounded domain subset of D containing

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Q: If vo and V, are respectively the t-conorms that are dual to the t-norms Ao and A5 respectively.…

A: In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary…

Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

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Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

A: Let A be a non-empty compact subset of ℝ Let B be a non-empty closed subset of ℝ such that A∩B=∅ To…

Q: Prove that T is the discrete topology for X iff every point in X is an open set

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Q: 5. Let X =[0,2). Define T = {[0,a):0<a<2}. %3D a) Show that T is a topology on X. b) What are the…

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Q: {X, Ø, {1}, {2},{1,2},{1,3}}, is (X, t) a connected a) Let X = {1,2,3} with the topology t=…

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Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: Let Y be an infinite subset of a compact set X CR. Prove Y' 0.

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Q: sketch the set {z ∈ C : Re(z) = Im(z + 2 + i)}

A: We have to sketch the set {z∈ℂ:Re(z)=Im(z+2+i)} Let z=x+iy where x,y∈ℝ

Q: Exercise 4.6. (1) Consider R² with the standard topology. Let p ER² be a point not in a closed set…

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Q: Let T₁ and ₂ be topologies on X. Show that J₁ J₂ is a topology on X.

A: Let us consider the two topologies, T" data-mce-style="display: inline-table; font-style: normal;…

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

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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: Let π D₁ = {z € C\{0} : < Arg z < 4 D₂ = {z € C: 0 ≤ Re(2) ≤ ½ & 0 ≤ Im(2) ≤ 1}. a) Sketch D₁ and D₂…

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Q: Consider the number line R with the topology t = {Ø, R, (-0, a); a E R}. Is the set A = (-3, 0]…

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Q: Let (X.J) be a topolgical space and {C1, C2 Cn} be a finite collection of compact subsets of X.…

A: Given Let (X,I) be a topological space and C1,C2,......Cn be a finite collection of compact subset…

Q: Prove that he interval E = (a, +oo) is an open set in R. (Show all details of your work).

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Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

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Q: Let V C(R ) be a nonempty open set satisfying a – b EV whenever a, b e V. Prove that V = R. %D

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Q: Classify each of the following subsets of R as compact, connected, neither, or both. Justify your…

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Q: 9. Describe and sketch the locus of z satisfying the condition |z – 1+ i| = |z – 2|

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Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

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Q: Suppose X is a topological space having only a finite number of components. Prove that each…

A: given that  suppose X is a topological space having only a finite number of components . we have to…