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

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A: τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normalspace or…

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A: Given:

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A: A matric space X is separable if it has a countable dense subset.

Q: If V is an inner product space and = for every u,ve V, which of the following is true? O A) u + w O…

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Q: - Let (X, t) be a topological space and A, BCX, then 1- A° 2 A- b(A) 2- A° # A- b(A) 3- A° = A –…

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Q: Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B)…

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Q: (a) Suppose that (M, d) is a metric space. Consider the mapping f: M [0, 1] be given and let di(x,…

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Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: Let (X.d) be a metric space, x ∈ X, epsilon > 0, and E = { y∈ X: d(x,y) ≤ epsilon}. Show that E is…

A: A set E is closed in X if it contains all of its limit points.

Q: let and be nonempty metric Space x E X. Ihen A B Sebsets of and let d(x, A), dlA, B) and dlA) are…

A: As per bartleby guidelines for more than three subparts in a question only first three should be…

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: Let (M, d) be a metric space. Show that the set {B(x, 1/n); x E M; n e N*} is a Basis for the metric…

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Q: Let (X, d) be a separable metric space. into l∞.

A: This is a problem of functional analysis.

Q: Let

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Q: Prove the following. Let J be any set, and let {Xa : a E J} be an indexed family of connected…

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Q: Theorem 2.28. Let A be a subset of a topological space X. Then Int(A), Bd(A), and Int(X – A) are…

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Q: Let (X, T) be a topological space and A, B be a subsets of X then: اختر احدى الدجابات int(A - B)=…

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Q: Let A be a non-empty subset of the metric space (X,d). show that |d(x, A)- d(y, A)| < d(x, y)For all…

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Q: 2. Find the complement CC of the set C with respect to the space C if (a) C = {x :0 < x < 1}, C = {x…

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Q: Let X = {a,b,c} prove that topological space on the set X T = {X. Ø, {b}, {a,b}} be a

A: Let X={a,b,c} be a set. To show that T={X,ϕ,{b},{a,b}} be a topological space on the set X.

Q: Given (Xi,d;), i = 1,n as a metric space and we define the set X = II-1X; =X;*X2*X;*...Xn %3D Show…

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Q: If M and N are metric spaces, the Cartesian product M x N is again a netric space with the distance…

A: We have given that , M and N are metric spaces. We have to show that , M × N is again a metric space…

Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

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Q: Let v e R" and letk e R. Prove that S = {x € R" :x•v = k} is an affine subset of R". %3D

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Q: let C be a compact subset in a Hausdorff topological space (X,T) and b∈X a point. If b∉C then prove…

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Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. biso (2) If…

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

Q: Let X be a topological space. Construct a topological space Px and a continuous surjection 0: X→ Px…

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Q: B) Let (R, T) be the left ray topological space If Y=(3,7], then define ty and determine the open…

A: Given: B) ℝ, τ is a left ray topological space and Y=(3, 7]. To determine: Which one of the open and…

Q: For every ,x, y E R, let p(x, y) =| x² – y² |. Is p a metric -

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Q: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

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Q: Let A be a non-empty subset of the metric space (X,d). show that a. d(x, A) – d(y, A)| < d(x, y)For…

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Q: Let {Xa | a E A} be a collection of Hausdorff topological spaces. Prove that IαEA Xa is Hausdorff…

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Q: Let A be a sSubset of a metric space A. Given r€A, define the distar from x to A to be dist(z, A) =…

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Q: Let X = {a.b.c.d.e) Find the topology r on X generated by f-{{a.b.c).(b.c.d).fc.d}}

A: X={a,b,c,d,e}

Q: Let E be a Euclidean space with inner ratic form. For all u, v E E, we have the C y(u, v)² ≤…

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Q: (2) Let (X,r)be a topological space, If for every A open set, and x E A", 3 Ux and Vy open sets,…

A: A topological space X is Hausdorff if for each pair of distinct points p and q in X , there are…

Q: Show that (C(X),|| · ||∞,x) is a metric space.

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Q: Let (X, d) be a metric space and let A be a non-empty subset of X. Prove that A is open if and only…

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Q: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

Q: Show the following result: Theorem. Let (X, d) be a metric space, A C X, and let u: A → R be an…

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Q: Let (X, d) be a compact metric space and T: XX be a mapping that satisfies d(Tr,Ty)<d(r,y) for all…

A: What is Metric Space: Metric space is an abstract set in mathematics, particularly topology, with a…

Q: Let X = R, and let T consist of 0, R, and all intervals of the form (a, b) where a, b e R and a < b.…

A: A topological space is an ordered pair {\displaystyle (X,\tau ),} where {\displaystyle X} is a set…

Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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Q: An Elliptical norm for R2 is defined by + 4u2, where x = (1,u). (i) Prove that l is a norm for R2…

A: This is a problem of Functional Analysis.

Q: Let E and F be compact sets in a metric space (X,d). Show that E U F is compact using the definition…

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Q: A° U B° c (A U B)

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Q: Let V be an inner product space. Then for x, y, z ∈V and c ∈F, the following statements are true.…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.