Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then Such topology doesn't exist O X is infinite O X can be either finite or infinite O X is finite

Q: In the discrete topology on the set X, every set is open and it also follows that every set is…

A:

Q: 2. Let X = {1,2, 3, 4} and let T = {ó, X, {1}, {3},{1,3}} be a topology on X, then {2, 4} is a. open…

A: Since you have asked multiple question , as per our guidelines we are supposed to answer only one…

Q: Let X = {1,2, 3, 4} and let T = {, X, {1}, {3},{1,3}} be a topology on X, then {2, 4} is a. open in…

A:

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: Prove the following: Let X denote the set {0,1} with the discrete topology. Then X" is homeomorphic…

A: Proof-) The Cantor set itself is a cantor space. But the canonical example of a cantor space is the…

Q: 2. (a) Use axiom (2) in the definition of a topology to show that if {U1,U2, . ,Un} is a finite…

A: It is given in part (a) that (X,t) be any topological space, and the interval (x-A,x+A) is only open…

Q: Let X be a topological space. Show that the following are equivalent. (i) X is disconnected. (ii) 3…

A: Let X be a topological space then, To prove: Following statements are equivalent: (i) X is…

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

A: To check: Whether the statement is true or false: Given: The statement is "The collection of all…

Q: Prove that ifS is a bounded set, then inf(S) = -sup(-S) %3D

A:

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

A:

Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Two discrete spaces are homeomorphic if there exists a bijection between them and it is possible if…

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: Use the Heine-Borel theorem to give an example of a set not compact in R with the usual topology…

A: An example of a set not compact in R with the usual topology and a non-compact set R2 with the usual…

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…

A:

Q: Let X be a non empty set and p be a fixed point in X. Define T={X, AcX such that pgA}. Then {p} is:…

A: Solution: Given that the topology is T=X, A⊂X such that p∉A

Q: Let X and Y be two discrete spaces, then * O X is never homeomorphic to Y X is homomorphic to Y if…

A: option (c) is correct.

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: Prove that if S is a bounded set then S is bounded.

A:

Q: Let X and Y be two discrete spaces, then! X is never homeomorphic to Y X is homeomorphic to Y if and…

A:

Q: Let X = {a, b, c}, T1 = {X, p, {a}, {b}, {a,b}} and T2 = {X, p, {a}, {c}, {a,c}}. Then one of the…

A: Given that set X={a,b,c}

Q: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, is a topology on X.

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

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

A:

Q: Theorem 2.1. Let{U}{_, be a finite collection of open sets in a topological space (X,T). Then N, U…

A:

Q: 7. Let X be an infinite set equipped with the finite closed topology. A finite subset of X is a.…

A: Option (a) is correct.

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: 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: Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set…

A: Given that A is a subset of the topological space X and p is a point in X. Consider a sequence…

Q: Let X = {1,2, 3, 4} and let T {ø, X, {1}, {3},{1,3}} be a topology on X, then {2, 4} is a. open in…

A: Our objective is to choose the correct answer

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

A:

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

Q: Let X and Y be two discrete spaces, then* O x is never homeomorphic to Y O X is homomorphic to Y if…

A: As two discrete spaces are homeomorphic iff they have same cardinality .

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

A:

Q: Let X be a set with more than one element. (X,7) is a topological space such that Vxe X, {x} €T then…

A: Given the set X contains more than one element.

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

A: Connected means it cannot be written as union of disjoint open sets

Q: consider R with the usual topology then a subset A={1,0.5,..} is: empty or closed or open or none…

A: Given R with the usual topology and given a subset  A=1, 12, 13,.....

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

A: An infinite set equipped with discrete topology is not compact.

Q: 7. Suppose that a set X is equipped with the finite closed topology. Which one of the followings is…

A: As per the policy, we are solving first question. Please repost the question and specify which…

Q: 4. Prove that the set Z is a closed set in R.

A:

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: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

A: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

Q: (2.2) Let C = {a, b, c, d} and T = {0, {a, b, c} , {b, c} , {a},C} be the topology defined on C. %3D…

A:

Q: 2. Let X = {1,2, 3, 4} and let T = {ø, X, {1}, {3}, {1,3}} be a topology on X, then {2, 4} is a.…

A: b. closed in (X,T)

Q: Let X be infinite set and x0 ∈ X. Consider the topology τ = {G ⊆ X : x0 ∈ G or G = ∅} on X. Is (X, τ…

A:

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 X = {a, b, c}, T1 = {X, 0, {a}, {b}, {a,b}} and T2 = {X, 6, {a}, {c}, {a,c}}. Then one of the…

A: According to experts guidelines of bartleby i have to solve only first problem so repost for further…

Q: Prove that a set A is connected iff A is and interval.

A: