Let X be an infinite set and I be a topology on X. If every infinite subset of X is in T, then Tis * cannot be a topology the discrete topology the Euclidean topology the finite closed topology

Q: Example. The standard topology Tstd on R" is defined as follows: a subset U of R" belongs to Ttd if…

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Q: If  X is a countable set show that every topology on X is second countable.

A: To Show: If  X is a countable set show that every topology on X is second countable.

Q: Let O be the collection of intervals Ia = (a, ∞) where a R along with I = 0 and I-∞ = R. Does this…

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Q: For any topological space (X, T) and A C X, the set A is closed. That is, for any set A in a…

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

Q: Let X be an infinite set equipped with the finite closed topology. A finite subset of X is a. closed…

A: The "finite closed topology" describes the closed sets. A subset U is open if and only if U=∅ or X∖U…

Q: Let X = {a, b, c, d, e, f, g} and B = {{a,c, e, f}, {b, c, d, f,g}, {c}, {f}}. Show that B is a base…

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Q: Let X be an infinite set equipped with the indiscrete topology T, then (X,T) is compact. * True…

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Q: 1) Let X= {a, b, c} List the open sets in the following topologies on X: (i) The discrete topology;…

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Q: Show that the dictionary order topology on the set R x R is the same as the product topology Rd × R,…

A: We have to show that the dictionary order topology on the set ℝ×ℝ is same as the product topology…

Q: Let T be the finite closed topology on a finite set X, then * O Tis the Euclidean topology O Tis the…

A: Thanks for the question :)And your upvote will be really appreciable ;)

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

A: Answer

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: 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: I? B is a family of subsets of X Such that: (2) If B, ,BzE/ß them B,nB, EB Then B is a base for a…

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Q: Let X be a topological space. Prove that (a) Ø and X are closed. (b) Intersections of any number of…

A: Union of any open sets is open and intersection of two open sets is open.

Q: Let X be an infinite set T be topology infinite subsets of that is on X- and let on X in which all…

A: Given :  X be an infinite set and τ be a topology on X such that all the infinite subsets of X are…

Q: Let X be an infinite set and T a topology on X. If every infinite subset of X is in T, prove that T…

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Q: Let X be an infinite set T be topology of the a Linfinite subsets that is on x- and let on X in…

A: Given : X be an infinite set and T be a topology on X such that all infinite subsets of X are open.…

Q: A non-empty set A in the standard topology on R so that ðA= 0.

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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. If {T,} is a family of topologies on X, show that Ta is a topology on X. Is UT, a topology on X?…

A: First we will prove the result that arbitrary intersection of a topologies is a topology by showing…

Q: 4) LetXbe any set with the discrete topology. Let Y be any subset of X. Prove that ô( Y ) =Ø.

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Q: Let X be an infinite set equipped with the indiscrete topology T, then (X,T) is compact. * True…

A: Note: We are entitled to solve only one question at a time. As the specified one is not mentioned,…

Q: If X #0, compare the following topologies on X: a) The co-finite topology 7, and the co-countable…

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

A: Which of the following statement is true?

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

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Q: Q1.Consider X ={a,b, c, d, f, g} .Let T={ó, X, {a,b, c} ,{g};{a,b, c, g}} (a) Prove that T is a…

A: Since you post a question with multy part, we solve only first three, if you need any particular…

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

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Q: if

A: R is not connected if T is the finite closed topology.

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

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Q: In a Cofinite topology of X the collection of subsets is: a) r = {AcX|A±Ø or A is finite} b) r = {Ac…

A: Let X be a non empty set , then the collection of subsets of X whose compliments are finite along…

Q: 100. In a topological space, which of the following statements is correct? (A) a compact set is…

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

A: Any two homeomorphic spaces always shares the same Topological property. This means if one of them…

Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

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Q: Ris not connected if T is the trivial (usual) topology O the lower limit topology the indiscrete…

A: We know that

Q: Let X be an infinite set and T be a topology on X. If every infinite subset of X is in T, then T is…

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Q: Ris not connected if T is * O None of the choices O the trivial (usual) topology the indiscrete…

A: R is not connected if T is

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

A: Answer

Q: 6) Let p and q be two points in a set, X. Consider the two collections of subsets of X: I. = {Ø, X,…

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Q: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then…

A: X is finite

Q: Let X be an infinite set and T be a topology on X. If every infinite subset of X is closed, then T…

A: Given that Let X be an infinite set and T be a topology on X.If every infinite subset of X is…

Q: Prove that a set S⊂R is everywhere dense iff it intersects every non-empty open sets in R. Prove…

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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}.…

A: We can show that X is countable. And by taking singleton we can solve the problem

Q: R is not connected if T is * the indiscrete topology the trivial (usual) topology the finite closed…

A: Option 3 is correct.

Q: Let X = {a, b, c}. Is the collection {X, Ø, {a}, {a, b}, {c}} a topology on X?

A: Here, the given set is X=a,b,c and collection is X,∅,a,a,b,c.

Q: Give an example of a set in R ^ 2 with the usual topology that does not have any subset (not…

A: Given that R2 with the usual topology. The aim is to find an example of the set  R2 with the usual…

Q: Let X be ametric space. If A and B are each dense open subsets of X then AOB is dense open subset of…

A: First we have shown the intersection is open. Then we use the property that if a set intersect every…

Q: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then* OX…

A: Gievn that Let T be the finite closed topology and be also the discrete topology on a non empty set…

Q: On a set X, consider the collection consisting of four of its subsets, given by Γ = {X, ∅, A, B},…

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