Question: Find a non-singleton element of the discrete topology on {0,1}×N which is not an element of the order topology on {0, 1} × N.

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: u* (B) = inf{l&l:s is a covering of B by closed boxes} for all BcR" صواب İhi

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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: Q3) Discuss these examples (a) Let X be an infinite set and t is the cofinite topology on X. Then…

A: We shall solve these problems in next step. Please read the answer carefully and slowly to…

Q: 4. Let E1, E2 be two compact subsets of R'. Use the Heine-Borel Theorem to prove that EUE2 is also a…

A: As you have asked more than one question, we Shall answer first one only. For others please post…

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: Q4) Consider the topology Y= {ACR:OGAJufo} on R ad A- RIF0}. Prove that CA is the discrete topology…

A: Given ζ=A⊂ℝ:0∈A∪∅ is a topology on ℝ. To show ζA is the discrete topology on A. By definition,…

Q: We define the included point topology by Tp={ UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R…

A: option (d) is correct.

Q: Suppose X has the discrete topology. Then the infinite product X" with the product topology is also…

A: Suppose X has the discrete topology. Then the infinite product Xw with the product topology is also…

Q: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection ?…

A: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection?…

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: Which of the following is not a base for any topology on R? a. {(a, b]: a,b ER and a <b} O b. {[a,…

A: Base for a topology: The base for topology of any set X is collection B of subsets of X which…

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

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Q: Lonsider the following subsets of R equipped with the Euclidean topology: 3n 1 A = { n+ 1 B = Ü 1+…

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

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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: 4 ,3 ,2 ,1 = X آقا -X L find The Possible Topology That Contain sus { 1123 , 2, 3, {\/ 3 ، 4 seas

A: It is given that,  X = {1, 2, 3, 4}. We have to find the possible Topology on X that contain the…

Q: The collection A-{(-,1),(-1,)} of subsets of Ris for the usual topology on R a topology a base and a…

A:

Q: Q₂/(a) Let X=(1, 4, 5,7, 8, 9) and o= {(1,4),(4,7, 8}}, show that whether o is: (1) a subbase for a…

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

Q: We define the included point topology by Tp={ UcR;U=ø or peU}. Let A = [3,5[, %3D then A is dense in…

A: See the detailed solution below.

Q: 5) We have seen that the collection of subsets of Z given by = {B(n) | B(n) = {n} when n is odd and…

A: I have just used the definition of basis and open sets in topology

Q: 6. If X is any set, the collection of all one-point subsets of X is a basis for the discrete…

A: This is the problem of topology.

Q: Which of the following is not a base for any topology on R? O {(-0,a): a E (-∞, 0]}U{(0, ∞)} O {(a,…

A: Base for a topology: Let X, τ be a topological space. A collection of open sets B is said to be base…

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

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Q: if

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

Q: We define the included point topology by Tp={ UcR;U=Ø or peU}. Let A = [3,5[, then A is dense in R…

A:

Q: on N and the floor topology on N are Exercise 3.42. Show that standard (order) topology the same.

A: Standard topology: Standard topology on ℝ is generated by the basis on usual open intervals B=a,…

Q: The collection {On}3, where On = (1/n, 1-1/n), is an open cover of the set A = [1/2,1). Find (a)…

A: We have to find a limn→+∞1-1n, b∪n=3+∞On, and c∩n=3+∞On

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: Q2. Answer as true or false: (a) f-'((AUB)nC) = (ƒ-'(A)n f-1C)) U (f-1(B) n f-1(C)). (b) Given the…

A: (a) is True. (b) is False.

Q: (d) Consider the set [1, √5] C [−1, √10]. Explain f is not continuous by connectedness, compactness,…

A: Given: The function, f:-1,10→ℝ defined by, fx=x2,x&gt;2x,x≤2 The set 1,5⊂-1,10. To explain: Why the…

Q: 24. The smallest topology on R containing the collection * {(-00, a) : a e Q} is : the left ray…

A:

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 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: We define the included point topology by Tp={ UcR;U=Ø or pEU). Let A = -[3,5[, then A is dense in…

A:

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

A: R is not connected if T is

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: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

A: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

Q: We define the included point topology by Tp%={UcR;U=Ø or pEU}. Let A = [3,5[, then A is dense in…

A: Given that the topology is defined as  Tp=U⊂R,U=∅ or p∈U. Let A=3,5. We say that A is dense in R if…

Q: (i) the finite complement topology on R is strictly coarser than the standard topology on R. (ii)…

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

A: Bone of choice is correct one

Q: If X is any set, the collection of all one-point subsets of X is a basis for the discrete topology…

A: Given : Let (X, T) is a discrete topological space on the set X. To prove : We have to show that all…

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: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

A: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

Q: u (B) = inf{lal: a is a covering of B by closed boxes} for all BCR" صواب İhi

A: True

Q: Let X = (1, 2, 3, 4) and T be the discrete topology on X. Consider the following statements: (a)…

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Q: Suppose X has the discrete topology. Then the infinite product X with the product topology is also…

A: It is given that X has the discrete topology. Check whether the infinite product Xw with product…