Exercise 2.46. Show that: (1) The topology generated by a subbasis is a topology. (2) The sets in the subbasis are open sets in that topology.

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: 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: Let T be the Euclidean topology. Let A = [0,2] and B = [0,1[, then B is clopen in (A, TA) where TA…

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Q: a. Show that Ø, R E T. (by definition of Topology)

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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…

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Q: 1. Let X = {a,b,c} and B={{a,c}.{b,c}}cP(X). Show that B cannot be a base for any topology r on X.

A: According to the answering guidelines, we shall solve first question only. If you want others to be…

Q: Show that Loo is not icomorphic to a subepace of

A: Consider the spaces l∞   and   lp where p∈1, ∞. It is known that l∞ is not a separable space but, lp…

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: A ° = A – JA

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

A: Answer

Q: 3.5 Show that the metric space (R, d), where d(r, y) = |x – y|, is separable.

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

Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology.   We claim that Z is NOT…

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

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Q: Find the image of the infinite strip <ys under the trans-

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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: Consider the set A=(x:x>a, a E R}U(x:xsb, bE R). Find the topology on R which has A as subbases.

A: Given :               Given a set                   A = x : x &gt; a, a∈R ∪ x : x ≤ b, b∈R     To…

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

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Q: Let 7 be the Euclidean topology. Let A [0,2] and B [0,1, then Bis dopen in (AT) where 7, is the…

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Q: 3 let X= {ab,c},B =} {n,b}, {9c3}.5how that Whether B is a %3D %3D base for any topology on X or not

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Q: Exercise 2.27. Show that a set U is open in a topological space X if and only if every point of U is…

A: Let, U is an open set in a topological space X. Then, U⊂U  So, U⊂int(U) and int(U)⊂U Hence, U=int(U)

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: 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: The quotient topology actually defines a topology.

A: The quotient topology  actually defines a topology

Q: Provide an example of a topological space that is connected but not path-connected,why?

A: We use definition of connected and path connected sets. We use contradiction method.

Q: Use only the definition of affine dependence to show that an indexed set {v1, v} in R" is affinely…

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Q: if

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

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

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Q: In R with the cofinite topology, every preopen set is open

A: As we know that , In ℝ with discrete topology of every semi open set is open.

Q: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

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

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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: Consider the spaceZ+with the finite complement topology. Is thisspace a Hausdorff space? Is this…

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Q: Lecture 8 Review Exercise: 1) What does it mean if we say that a set S is closed under a binary…

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

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

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: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

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Q: Consider X = {1,2,3, 4, 5} where the topology is {U C X|1 € U}. Determine whether or not X is…

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Q: Amongst the various possible products of four topological spaces A, B, C and D, show that (A × (B ×…

A: We shall solve this in next step

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: 2. Let T be the cofinite topology on R, and let A = (-x, 1) U (1, ), B = (1,2). Fine the boundary…

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Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: Let T be the Euclidean topology. Let A = [0,2] and B = [0,1[, then B is clopen in (A,T) where T is…

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

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…

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Q: Let T = {X. Ø, {b}, {a, b}} be a topology on X = {a,b,c} and let A = {a,b,c}, B = {a,b,c}. Find a)…

A: Given That: Let T={X,ϕ,{b},{a,b}} be a Topology on X={a,b,c} A={a,b,c} ,B={a,b,c} To Find: 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