Prove that the boundary points of an open interval (a, b) CR are the points a and b.

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: The set of all directed line segments that are equivalent to a given directed line segment PQ\is…

A: The set of all directed line segments that are equivalent to a given directed line segment PQ \ is a…

Q: Let A and B be subsets of topological spaces X and Y, respectively. Prove that CI(A × B) Cl(A) ×…

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Q: A(n)................................... is the collection of all points in the plane such that the…

A: We know, A parabola is a collection of all the points in a plane such that the distance from each…

Q: Consider the reglon bounded by the graphs of y- and y- VE

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Q: Suppose A ⊂ R is both closed and bounded, prove that A has the Heine-Borel Property

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Q: Suppose that l is a line. Let H be one of the two half-planes bounded by l. Give a proof that l U H…

A: Answer is mentioned below

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 X be any set with the discrete topology. Show that X is normal.

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Q: Let (X, d) be a metric space and let A ⊆ X be complete. Show that A is closed.

A: Given that X,d be a metric space and A⊆X be complete. The objective is to show that A is closed.

Q: Show without using the Heine-Borel theorem that the line R is not a compact set in R ^ 2

A: The proof is done using the definition of compact set. A set S is said to be compact if and only if…

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 ° = A – JA

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Q: Find the limit point of the open interval (a, b) in IR, where a

A: We have to find the limit point of the open interval a, b in ℝ  where a<b. Limit point: Let S be…

Q: Show that X is Hausdorff if and only if the diagonal △ = {(x, x)|x ∈ X} is closed in X × X.

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Q: Consider the closed curve C, which is th

A: Given, the closed curve C, which is the image of the path…

Q: Determine whether is a linearly independent set of vec- 3 11 tors in R3.

A: A set of vectors v1,v2,v3,…,vk in a vector space is said to be linearly independent if…

Q: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

A: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

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

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Q: If M is a closed linear subs r space X, and x, be a vect the distance from x, to M, ional f, in X'…

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Q: Show that the closed interval [0, 1] and the unit circle S' = {(x, y)|x² + y² = 1} cannot be…

A: Suppose there exist a homeomorphism f:[0, ∞)----->R then restriction of f f|[0,∞)\{0} : [0,…

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

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Q: Let R have its usual topology and let X = {a,b, c}. Define f : R –→ X by а, х

A: Given, ℝ have its usual topology and let X=a, b, c. Define f: ℝ→X   by…

Q: Let (X,d) be any metric space, then the only open sets and closed sets are and X O True O False

A: Given statement is false because if we take descrete metric space

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

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Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

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

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Q: we diveagena theosem to eveluate ty3j + XzK. dA ahose sethe boundary Of cube defined by

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Q: Let (X, M, u) be a measure space wE

A: Given, (X, Μ, μ) be a measure space where μ is a probability measure. Assume thatf, g : X→ℝ are…

Q: 1. Show that the real line is a metric space.

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Q: Consider the number line R with the topology t = {Ø, R, (-0, a); a E R}. Is the set A = (-3, 0]…

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Q: Compare the vertical interval topology with the standard topology on R?. Which one is finer, the…

A: The

Q: Show that the linear map (x, y)H (x+ y, x- y) from (R?, || - |1,) to (R?, I| - ||is an isometry.

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Q: Q7: Prove that ( IR", ||·|| ) is complete metric space.

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Q: 1) Prove that for any space, X, with any topology, T, that: (i) CI(X) = X and (ii) Int(X) = X.

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Q: Prove or disprove that AS is congruent to SB (or that CS divides AB equally) if a line is drawn from…

A: Given, in △ABC, AB=AC=aBC=bBD=CDAD=h (altitude) To prove: CS does not divides AB in two equal parts.…

Q: Let X be an uncountable set and let T = {U C X :U = Ø or U¢ is finite} %3D Then is topological space…

A: let X be an uncountable set and T=U⊑X : U=∅ or Uc is finite i.e cofinite topology on X. Then the…

Q: Use the Heine-Borel theorem to show that every closed and bounded interval [a, b] with the absolute…

A: Compact Set: A collection A of subsets of space X is said to be cover X or covering of X if the…

Q: Assume X is Hausdorff. If K C X is compact and x ¢ K, show that there exist disjoint open sets U and…

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Q: - Consider noncoplanar points A, B, C, and D. Using three points at a time (such as A, B, and C),…

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Q: Give all boundary points of D.  Is D open, closed or bounded? Justify all your answers.

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Q: Consider the topology 7 = {GCR:0 ¢G or G = R} on R. Is open interval (-1, 1) compact in (R, T)?…

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

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Q: Exhibit an open cover of the interval (1, 2] that has no finite subcover.

A: An open cover of A is a collection of open subsets of A whose union is all of A and that a finite…

Q: Given, we have a finite purely non-atomic measure space (X, F, µ), prove that µ(F) = [0 µ(X)].

A: Given: Here, the given is finite purely non-atomic measure space (X,F,μ). To prove that μ(F)=0, μ(X)

Q: 2. Does d(x, y)= (x-y)? define a metric on the set of all real numbers?

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Q: Let x1 and x2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: Given: G is midpoint of IE 21 = 2 Prove: ATIG = AREG 2 1. 3,

A: Given that, G is the midpoint of IE¯ So IG¯=GE¯ ∠I=∠E∠1=∠2

Q: Show that the real number R with d(x,y)=|x-2y| are not a metric space

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Q: A° U B° c (A U B)

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