Question 7. Let (X,r) be a topological space and A, BCX, Determine whether A-B2A-B or not.

Q: Question 6. Let (X,7) be a topological space and A, BC X. Determine whether (A- B) CA -B or not.

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

Q: QUESTION 8 The set M with the usual metric is connected. 2 O True O False

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Question 9. Prove or disprove. If (X,7,) and (X,r,) are T,-spaces, then (X,7,nt,) is also a T -…

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Q: Let X = (1, 2, 3), Tx = (0, X, (1), (2, 3}}, then (X, Tx) is %3D %3D 1) a T1 space 2) a T2 space

A: T1 space: A topological space X has the T1 property if x and y are distinct points of X, there…

Q: Question 2. Let (X,r) be a topological space and let A be a subset of X. Determine whether the…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

Q: QUESTION 5 The set R-{0} with the usual metric is connected. O True O False

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Q: Let (x,d) be metric space and A, BC X. 2. Show that (AnB)= A'n B and AUB = AUB %3D

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Q: Let X and Y be two T, spaces, then X×Y. :is

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Q: Question 1: Show that X is Hausdorff if and only if the set S = {(x,x): x € X} is closed in X X X.…

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Q: Question 4 . Let (X, d) be a metric space and let p(r, y) is complete if and only if (X, p) is…

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Q: 3. Let (X, T) be the topological space defined in question 2., then the closure of {2} is a. X b.…

A: It depends on the given topological space. Here topological space is not specified. If it's usual…

Q: Q. Assume that (X,T) is a topological space. If A = {m, n, p} CX is a closed set. Then one of the…

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Q: QUESTION 8 The set M={XEN:x<} with the usual metric is connected. O True O False

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Q: QUESTION 5 The set R- {0} with the usual metric is connected. O True False QUESTION 6 The set {0,3}…

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Q: Question 2. Let (X,r) be a topological space and let A be a subset of X. Determine whether the…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

Q: Question 3. Consider the topology 1={X,.{a),{a,c),{a,b,d}} X = (a, b,c,d). Let A = {a,d}, find A and…

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Q: Question 4. Let A be a subset of a topological space (X,r). Prove that the following statements are…

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Q: Theorem 2.55. Let X and Y be spaces, A c Х, В С Ү, апd give X x Y the product topology. Then Aх В %3…

A: Product topology :Let X,Tx and Y,TY are two topological spaces, then product topology on X and Y is…

Q: O Let ( X, T) be topdogical space and A,B CX- ldentify the Condition when a (Int(A)u FrlB) Ul F CA)…

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Q: Question 1. Let ACX. Show that 1= {UcX:AcU}U{®} is a topology on X.

A: For a set X having certain collection of subsets represented by τ will be a topology on X if it…

Q: Question 1. Let (X,7) be a topological space and let F be the collection of all closed sets, Show…

A: Closed Sets: A subset of a topological space is said to be closed if the set is open De Morgan's…

Q: QUESTION 9 Suppose metric space X is not path-wise connected then X is not connected. True False

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Q: QUESTION 8 The set M= EN: X< with the usual metric is connected. O True O False

A: Since you have asked more than one question. So we will solve first one for you. If you want other…

Q: Theorem 7.33. Let X and Y be topological spaces. The product topology on X × Y is the coarsest…

A: According to the given information,

Q: Question 9. Prove or disprove. If (X,7,) and (X,r,) are T, -spaces, then (X,7,nt,) is also a T, -…

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Q: Let [a, b] be an inter a space. Let ƒ:[a,b] → X

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Q: Q # 3(b) Let (G, H) be a disconnection of a topological space (X, 7), ant let AC X. If A is…

A: Before going to this proof, we prove the below statement which in turn proves the above result.…

Q: • Problem 1. Let (V, (-|·)) be an inner product space. Show, for all v, w E V, that |(v\w)| <…

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Q: Question 4. Let f:X→Y be a mapping from a topological space (X,7x) a topological space (Y,ry). Prove…

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Q: Question 6. Let (X,r) be a topological space and A, BCX. Determine whether (A-B)CA -B or not.

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Q: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: Theorem 9.13. Let (X, d) and (Y,e) be metric spaces. Then X × Y is a metric space.

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Q: Question 1. Let (x.r) be a topological space and let A be a subset of X Determine whether the…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

Q: QUESTION 5 The set R/{O1 with the usual metric is not connected. O True O False

A: 5. TRUE The set, R - {0} with the usual metric can be written as the disjoint union of, (-infinity,…

Q: Problem 1. Let M = span 2 -3 What is the dimension of the linear space (M, R)? 2,

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Q: Question 1 Suppose X and Y are metric spaces and f: X -> Y is an isometric bijection. Then X is…

A: Given: Suppose X and Y are metric spaces and f:x→y is an isometric bisection Thus x is complete if y…

Q: • Problem 1. Let (V, (-|-)) be an inner product space. Show, for all v, w e V, that |(v\w)| <…

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Q: Problem 4. Prove that a continuous surjection from a compact space to a Hausdorff space is a…

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Q: Theorem 7.2. LetX,Y be topological spaces and yo E Y. The constant map f : X → Y defined by f(x) =…

A: Given, X, Y be topological spaces and y0∈Y. The constant map f:X→Y defined by fx=y0 Shows in the…

Q: Question 4 . Let (X,d) be a metric space and let p(x, y)=+d(x, y) d(r,y) for all r, y € X. Show that…

A: complete metric space is a metric space in which every Cauchy sequence is convergent. In other…

Q: Question (6): verse is not true. Contractible space simply connected space. But con-

A: A manifold X is Contactible if its identity map is homotopic to some constant map.   A manifold X…

Q: Theorem 7.32. Let X and Y be topological spaces. The projection maps tx,Ty on X×Y are continuous,…

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Q: Let X=R and T = {]a, +oo[/a R}U{o,R}, then (X, 7) is a regular space. True False

A: This is a problem of topology.

Q: QUESTION 3 Any cover of a metric space is open. O True O False

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Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

Q: QUESTION 5 The set R-{0} with the usual metric is connected.

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Q: Question 7. Let (x,r) topological space and A, BcX. Determine be a whether A-B2A-B or not.

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Q: Question 1. Let (X,r) be a topological space and let F be the collection of all closed sets, Show…

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Q: Question 6. Let (X,r) be a topological space and A, BCX. Determine whether (A-B)' C A' – B* or not.

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Q: Problem 1. Let (V, {-|·)) be an inner product space. Show, for all v, w e V, that |(v\w)| <…

A: Given: V,·|· is an inner product space.   To show: For all v,w∈V, v|w≤v|v12w|w12

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.