2. Define a cofinite topology J on an infinite set X. Answer each of the following for (X, J): (i) Verify that T is a topology on X; (ii) Show that (X, J) is T₁, but not T2; (iii) Define the concepts of a (a)cover; (b) an open cover; (c) a sub- cover and (d) a finite cover of a set X in a topological space (X, T). Show that the space (X, J), an infinite set with the cofinite topology J, is compact; (iv) Suppose that A is a finite subset of X. Discuss the nature of the subspace (A, JA).

Q: 5) Let Xbe an arbitrary space and let J, and I be two topologies on 1 2 it. (i) Is T, U T always a…

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

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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: * Let R be with the indiscrete topology. If A = {1,3, 5, 7, . }, then A' = A O N O RO

A: This question is related to topology, we will use the definition of indiscrete topology to solve it.

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Q: A and B are open subsets of X with Cl(A) = X = Cl(B).

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Q: Let X={a,b,c} and Y={1,2,3} and t={Y,0,{1,2}}. The function f={(a,1),(b,3),(c,2)}. Then the topology…

A: X = { a, b, c } , Y = { 1, 2, 3 } T = { Y, ∅,  {1, 2} } f = { (a, 1), (b, 3), (c, 3) }

Q: Prove the following sets are not compact by finding an open cover that does not have a finite…

A: a) Given-  A set A=−1,1. Explanation- Consider a set Xn=-2,1-1n for n=1,2.... Then clearly,…

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Q: (e) A = (-1,1)U{2} in the standard topology on R %3D (f) A = (-1,1) U {2} in the lower limit…

A: Definition:(1) Limit points: Let (X, τ) be a topological space, a∈X is the limit pointof the set…

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Q: Define a cofinite topology T on an infinite set X. Answer each of the following for (X, T):

A: Topology and cofinite topology definitions are recalled.

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