Let T be the finite closed topology on a finite set X, then * O Tis the Euclidean topology O Tis the discrete topology O X should be infinite T is not equivalent to any other topology ООО О

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

A:

Q: Let (R,r) be topological space with usual topology on R, then the set Qof rationals is O Neither…

A: ℚ is the set of rational numbers. ℝ is the set of real numbers. So, ℚ is a subset of ℝ .        i.e.…

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: 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 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: 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: 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: (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…

A:

Q: Select all that apply: The set II-1(-j,j) as a subset of X = R" is open in - The Discrete Topology…

A: In the given question we have to tell about the set s s.t why the set  is open in Discrete topology…

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

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Q: Show in the discrete metric space: 1. All sets are open (especially single points); 2. All…

A: We first define Discrete space

Q: 2. Let X = {a, b, c, d, e, f}. Which of the following collections of subsets of X is a topology on…

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Q: The collection of all closed intervals forms.( a subbase for a topology on R. This :topology is…

A: The collection of all closed intervalsfrom a subbase fo a toplolgy on RThis Topology is :-consider…

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: Let 7 be the Euclidean topology on R and 71 be the discrete topology on R. Define f : (IR, 71) → (R,…

A:

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: Prove that T is the discrete topology for X iff every point in X is an open set

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

A:

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: 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: Ler be a topology on X. Assume that is Husdouff and let xe X. 1. Show that {x} (and hence every…

A:

Q: Let X be an infinite set, then any topology on X is also infinite. Select one: O True O False

A: Note: " Since you have asked multiple question. As per our guidelines we are supposed to solve only…

Q: if

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

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

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

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

A: We know that

Q: (a) If {Ta} is a family of topologies on X, show that N Ta is a topology on X. Is UTa a topology on…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Let X = {a, b, c}, T1 = {X, 0, {a}, {b}, {a,b}} and %3D T2 = {X, p, {a}, {c}, {a,c}}. Then one of…

A: let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:   a) Both…

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…

A:

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…

A:

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

A: Answer

Q: 5) Let 7 be the euclidean topology on R and 7' be the finite-closed topology on R. Define f : (R,…

A:

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

A:

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

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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…

A:

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: 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: (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…

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

A: Every indiscrete space is connected.  as the union of two disjoint non-empty sets, so R is a…

Q: 4. Let X= {a,b,c} with the topology r ={X,ø,{a},{b},{c},}{a,b},{a,c},{b,c}}. Verify whether X is…

A: Given data X=a,b,c with the topology,  τ=X,ϕ,a,b,c,a,b,a,c,b,c

Q: ndary of a path-connected set in R^2 with the usual

A: We have to check that boundary of a path connected set in R2 with the usual topology is…

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: 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…