Ris not connected if T is * O the indiscrete topology O None of the choices O the trivial (usual) topology O 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 a. open…

A: Since you have asked multiple question , as per our guidelines we are supposed to answer only one…

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: 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: Q) Let 8= R and ZEfusR3Cup U23 Prove on R. or disprove that is a topology.

A: The set is defined as, τ=U⊂ℝ : 3∈U∪∅ 1) The first condition for topology is the set contains empty…

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: b) Let be a set of integers and T={o}{U,,ne N), on N. Show that whether (N,T) is 1) T₁-spr.ce or not…

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Q: Consider RxR to be a universal set with subsets A, B, and C defined as follows. A = {(x,y)\x² +y°…

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

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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 R if…

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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 R…

A: R is equipped with Tp and p=3

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

A:

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: al let (xt) betopology space and BCY then B is closed in y iff there is closed set CinX st B-COY

A: Let (X,Γ) be topology space and B⊂Y To Prove: B is closed in Y iff there is a closed set C in X s.t…

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

A: The detailed explanation is given below:

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: 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.

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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…

A:

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: (4.2) Let X = {1,2, 3, 4} and 7 = {0, {1,2,3} , {2, 3},{1},X} be the topology defined on X. (a) Show…

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

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Q: ket (X, @) be a topokgical space, Ka compact subset of X and (FalneN a family in PCX) with Fr+ Ø POR…

A:

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…

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: Which characteristic/s implies(y) that a et S of real numbers is closed? S has no accumulation…

A:

Q: None of the choices. Consider the following subsets of R equipped with the Euclidean topology: * Зт…

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Q: chow. that there ks no sset which har the set Q os its only. set of accumulation points

A: Before entering to the problem , first of all think about accumulation point. A point 'x' is said to…

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

A: Any two homeomorphic spaces always shares the same Topological property. This means if one of them…

Q: The collection of all left rays and right rays forms a subbase for a topology on R. This :topology…

A: Given , The collection of all left rays &right rays forms  a subbase for a topology on R.

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

A:

Q: 3. If ACX such that A# 0 and r (GCX: GnA = 0}U{X} then prove that r is a topology on X. If A {p},…

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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 *…

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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: 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: R is not connected if T is * the indiscrete topology the trivial (usual) topology the finite closed…

A: Option 3 is correct.

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…

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: 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: 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: 2. Let T={USR: U = Ø or U= R or U=(-∞o,a) for some a E I}. Prove that T is a topology on R (set of…

A: