Let X be an infinite set and T a topology on X. If every infinite subset of X is in T, prove that T is the discrete topology. Q2

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…

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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…

Q: Let X be any set with the discrete topology. Show that X is normal.

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Q: Verify that the open intervals on the line generate the Euclidean topology of the line. For this it…

A: We are using the theorem ,Union of open sets is always an open set to prove the above proposition.

Q: • A is open in X. • A = Int A • A contains none of its boundary points. • Every point of A has a…

A: As per the company rule, we are supposed to solve one problem. Therefore kindly repost the remaining…

Q: let x be an infinite set and let ta be a topology on x in which all infinite subsets of x are open…

A: Given that X be an infinite set and let τ be a topology on X in which all infinite subsets of X are…

Q: Which one of the following statements is true?

A: As in R with euclidean and R with finite closed topology then function f may not be continuous, thus…

Q: Let A be any set with the discrete topology. Show that A is normal.

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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: A set X and a topology on X which is not Hausdorff.

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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: Prove the statement: Let A be a subset of a nonempty finite set U. If P(A)=a, then P(Ā)=1-a.

A: We have to prove given statement...

Q: In this question let W := Mat(n, R) denote the vector space of real n x n matrices with n ≥ 2, and…

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Q: Give the definition of a Lebesgue measurable set. Prove that if A and B are Lebesgue measurable…

A: We are given that A and B are Lebesgue measurable sets.        => m(A) < ∞,            m(B)…

Q: Let S be a set defined as S={x∈R∣−1≤x≤1}. Determine whether S is open, closed, both, or neither in…

A: In the standard topology on R, a set is considered:Open if for every point x in the set, there…

Q: xercise 2. Show that a set with the discrete topology is compact if and only if it is finite. if and…

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Q: Which one of the following statements is true? * R with the Euclidean topology and R with the…

A: Two homeomorphic spaces share the same Topological property If one of them is compact, then the…

Q: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

Q: b Let A be a continuous fuzzy quantity. If all the a-cuts of A are closed intervals then show that A…

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Q: Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact. * O…

A: "Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact." The…

Q: Prove that P(Sn T) P(S) n P(T) for any sets S and T

A: Let, S and T be a any set. also, P(S) be any power set of S and P(T) be any power set of T.x∈P(S∩T)…

Q: Which one of the following statements is true?

A: In the R with the Euclidean topology and R with the finite closed topology, f might not be…

Q: 3. Let X be the set of positive integers. For each n E X, let Sn = {k € X: k2n}. Show that T = {S:…

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Q: Let X be a set and {Ya|a E A} an indexed family of sets. Prove that X U (Naea Ya) =Naea (X U Ya).

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Q: 1) Let Xbe an infinite set with the Finite Complement Topology. Let A EX be an infinite subset.…

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Q: If A and B are bounded subsets of R, then AuRand AOR are bounded: prove that

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Q: Show that if a topological space has a finite number of points each of which is closed then it has…

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Q: Cardinality: Let A and B be sets. We write A B (and say A has the same cardinality as B) if there is…

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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…

Q: Which one of the following statements is true? * Rwith the Euclidean topology and R with the finite…

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Q: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

A: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

Q: 4. If X #0, compare the following topologies on X: a) The co-finite topology Tf and the co-countable…

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