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

Q: Is R, equipped with the finite-closed topology, Hausdorff? Provide a proof supporting your answer.

A: Given: ℝ is equipped with the finite-closed topology. Let, ℝ with this topology be Hausdorff and x,…

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: 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: proof R with the finite-complement topology (F CT) is not continous to R with the usual topology.

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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…

A: We have to determine the following statements are true or false : i) Let X be an infinite set…

Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

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

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

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Q: Let A be a subset of a topological space X. Show that bd(A) = Ø iff A is open and closed in X.

A: We need to prove the statement.

Q: Use the definition of compactness to prove that a closed subset Y of a compact set X ⊆ R is compact

A: In this question, we use the definition of compactness to prove that a closed subset Y of a compact…

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: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify tha the quotient…

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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: let C be a compact subset Tn a Haus dorft topolopfcal space (K,R) and bex a If bac then prove that…

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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: Prove that every finite T,- space has the discrete topology.

A: T=(X, τ)  (X, τ) is a T1 space if and only if all points of X are closed in T.   Let X≠∅ be a set.…

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: Let X = {a,b,c} prove that topological space on the set X T = {X. Ø, {b}, {a,b}} be a

A: Let X={a,b,c} be a set. To show that T={X,ϕ,{b},{a,b}} be a topological space on the set X.

Q: Consider the set A=(x:x>a, a ER)U(x:xsb, bER). Find the topology on R which has A as subbases.

A: given a set  A=x:x>a,a∈R∪x:x≤b,b∈R to find the topology on R which has A  as subbases

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 a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: The objective is to choose the correct option: Let X be a discrete space then  a) X is homeomorphic…

Q: Let C(X,Y) have the compact-open topology. Show that if Y is Hausdorff, then C(X, Y) is Hausdorff

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Q: let C be a compact subset in a Hausdorff topological space (X,T) and b∈X a point. If b∉C then prove…

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Q: a) Consider the set X = {1,2,3} with the topology T = {Ø, X,{1}, {2, 3}, {1,2, 3}}. Show that (X, t)…

A: As per our guidelines we are supposed to answer only one asked question.Kindly repost other…

Q: Let T be a topology on X. Assume that T is Hausdorff and let x € X. (1) Show that {x} (and hence…

A: let T be a topology on x. T is a Hausdorff space

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

A: Topology Question

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

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

A: Answer

Q: Let Z be with the co-finite topology If A = {1,3,5, 7,.. }, then A A O O O

A: As Z is with co-finite topology. Every open neighborhood of an infinite set must contain infinitely…

Q: Prove the following: Let X denote the set {0,1} with the discrete topology. Let Y = X" = || X; ieZ+…

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Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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Q: a) Consider the set X = {1,2,3} with the topology t= (0,X, (1). (2, 3}. (1, 2, 3} }. Show that (X,r)…

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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 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: Show that the dictionary order topology on the set R × R is the same as the product topology ℝ_d × ℝ…

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

A: Given that X is  an infinite set with the countable closed topology T =(S subset of X: X-S is…

Q: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

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

A: Which one of the following statement is true

Q: If A is a compact subset of a metric space (X, d) and B is a closed subset of A, prove that B is…

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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* OX…

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

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

A: Given that X is an infinite set with the countable closed topology T-(S subset of X; X-S is…