4. Let X be any set and 7 be the countable complement topology on X. b) If X is uncountable, then show that (X, 7) is not a T;–space, for all i = 3, 4.

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: 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 = {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…

A:

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

A: This question is related to topology, we will use the definition of indiscrete topology to solve it.

Q: Let R be equipped with the Euclidean topology T and let Y =]10,20[. We denote by Ty the induced…

A: According to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Exercise 2.46. Show that: (1) The topology generated by a subbasis is a topology. (2) The sets in…

A: 2.46

Q: 3- Let X = {a, b} write a large Topology on X. 4- If A CX, find A' for (X.I).

A: (3) It is given that,  X = a, b. Then we have to write a large Topology on X. (4) It is given that,…

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

A:

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 (X, I) be a topological space and suppose that A and B are subsets of X such that ACB. Prove…

A: Let x ∈A°. Then by definition, there exists an open set U  such that  x∈U⊆A.Since A⊆B hence x∈U⊆B ,…

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: 7. Let X = {a,b,c}. Show that Bß ={{a,b},{b,c}}cannot be a base for any topology on Х.

A:

Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: See the detailed solution below.

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 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: Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset of X such that…

A: Let C be a connected subset of the topological space X.Also given that D is a subset of X such that…

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: Q3. Let N = {1, 2, 3, ..} equipped with the cofinite topology. Prove that M = {7,14, 21} is a closed…

A:

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: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

A:

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 R be equipped with the Euclidean topology T and let Y =]10,2O[. We denote by Ty the induced…

A:

Q: if

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

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 %3D a.…

A: Given X = 1 , 2 , 3 , 4 and T = ∅ , X , {1} , {3} , {1 , 3} is a topology on X Then open sets of (X…

Q: If X = {a,b, c}, let T1 = {0, X, {a}, {a,b}} and T2 = {0, X, {a}, {b, c}} . Find the smallest…

A:

Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A:

Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of 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: * Let R be with the co-finite topology. If A = {1,3,5, 7,..}, then Aº R O Q O A O N O

A:

Q: Q3. Let N = {1, 2, 3, ...} equipped with the cofinite topology. Prove that M = {7,14, 21} is a…

A: Given below the detailed solution

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

A:

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

A:

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)…

A:

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: (b) A = (0, 1] in the finite-complement topology on R. (c) A= {a, c} in X = {a, b, c} with topology…

A: The interior of a subset S of a topological space X is the union of all subsets of S that are open…

Q: 2- In a metric space (X, d), A is the intersection of all closed subsets of X which contains A.

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: Theorem 2.20. For any set A in a topological space X, the closure of A equals the inter- section of…

A: Let A  be a set in the topological space X. To prove that the closure of A  is the intersection of…

Q: Consider X = {a1, a2, ..., an} Can a topology of X with 3 open and one with 4 open be equivalent?

A: We have to verify the given statement.

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: 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 = {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 R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: Induced topology

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…