1. Let X be a topological space. Suppose X contains an infinite, closed, discrete subspace. Show that X is not compact.

Q: Show that [0, ∞) and (0, 0) ad subspaces of R with the usual topology are not homeomorphic.

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Q: Let X = {x, y, z} and B = {{x, z}, {x}, {y}}. Let T be the topology generated by B and consider B =…

A: Given: T is topology generated by B=x, z, x, y and B=x, z is a subspace of a topological space (X,…

Q: 1. Let A be a subset of normed space X. show that [4] is smallest closed subspace of X contain A

A: Statement:- Let Abe a subset of normed space X. Show that A is smallest closed subspace of X contain…

Q: The subset of pairs {(x, y) E R²|x,Y are integers } in the vector space R².

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Q: Let X be a locally compact Hausdorff space which is not compact; Let X* be one-point compactificatin…

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Q: 3. Show that closed subspace of a complete metric space is complete. 4. Let f be a continuous…

A: As per norms, the first question is answered: To prove that a closed subspace Y of a complete metric…

Q: Show that [0,∞) and (0,∞) as subspaces of R with the usual topology are not homeomorphic.

A: Given: [0, ∞) and (0, ∞) are subspaces of ℝ. To show: [0, ∞) and (0, ∞)are not homeomorphic.

Q: 3. Consider the set C(R) = {continuous functions of real numbers and consider its subsets. Show that…

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Q: Must a subspace of a hausdorff space be hausdorff? If it must be hausdorff then prove it. If it need…

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Q: Suppose M is an n-manifold with boundary. Show that ∂M with the subspace topology is an n −…

A: To prove that the boundary of a n-manifold is a (n-1) manifold without boundary

Q: (b) For subspaces X and Y, the sum X + Y is a direct sum (denoted X + Y) if every VEX + Y can be…

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Q: Suppose W,, W2,. (a) span (W, W2 (b) If S; spans W; for i = 1,...,r, then S, US, U...US, spans W, +…

A: (a) W1+W2W1+W2 would be W2W2. Below is the justification.  x+y∈W1+W2,wx+y∈W1+W2where x∈W1x∈W1 and…

Q: Let V be a 8 dimensional vector space over the set of reals. Then the dimension of a subspace of V…

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Q: Let L be a straight line in the Cartesian plane. Fully describe the topol- ogy that L inherits as a…

A: Let L be a straight line in the plane.  We have to find out the topology which L inherits as a…

Q: 6. Show that the closure Ỹ of a subspace Y of a normed space X is again a vector subspace.

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Q: 3. Use the e, d definition to show that f(x) = x² is uniformly continuous on A = [0,3]. %3D

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Q: Let Y be a closed subspace of a normed linear space X. Show that X is complete X are complete. Y AY…

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Q: If X is vector space and (X,d) is discrete metric space, then X can induce an inner product space…

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Q: Is not closed set in X x X. Then the space X is: (a) Hausdorff Space (b) Not Hausdorff Space (c)…

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Q: Show that in the finite complement topology on R every subspace is compact. Is R compact in the…

A: Recall that, a set  is compact if and only if every open cover has a finite subcover.

Q: (a) Let F be a finite dimensional subspace of an inner product space X. Then show that X=F+ F-and F…

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Q: - If X is a vector space and if X= MN, where M and N are subspaces, prove that N is isomorphic to…

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Q: Explain why the following statement is false:                          “A subspace is any subset of…

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Q: 61. Let Y be a subspace of X. A separation of Y is a pair of disjoint nonempty sets A and B whose…

A: Let Y be subspace of X. Here , A separation of Y is pair of disjoint nonempty sets A and B whose…

Q: Determine whether the given set S is a subspace of the vector space V. A. V=P3, and S is the subset…

A: We need to determine that given subsets are subspaces of corresponding vector spaces.We know that a…

Q: Let A CR be the set of integers, viewed as a subspace of the reals, and let X be the quotient space…

A: please find the attached image

Q: 2.57 † Let S = functions on [a,b]. Show that if n e N and if ao, a1,..., an are not all 0, then it…

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Q: If V is infinite-dimensional and S-is an infinite-dimensional subspace, must the dimension of V/S be…

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Q: Use F.Riesz Lemma to show that the closed unit ball in an infinite dimensional Banach space is not…

A: Let us prove above result

Q: 1 = { x:= = < |x|<1} am an 1. Consider the set Y=[-1,1] as a subspace of R. Is A=< open set in Y? Is…

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Q: f X is an arbitrary topological space, then we have the following: (a) each point in X

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Q: If W, and W, are two subspaces of a vector space V (F) then prove that WinW, is also a subspace of V…

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Q: Let (R, Tus) be the real line.Let r ∈ R such that 0 ≤ r < 1 and consider I = [0, 1] as a subspace of…

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Q: Show that a finite union of compact subspaces of a topological spaces X is compact-

A: We have to show that the finite union of two compact subspaces of a topological space X is compact.

Q: 5. If W is subspace, if A = : a. (A + B) is subspace these [8] € W and B = Non of above [8] € W,…

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Q: مممممم م العمل لها -Pre-Hilbert space - Metric space and Complete - Normed space and complet

A: 1) the banach space is solution definition of Banach space A Banach space is a normed linear space…

Q: Show that [0,1] and (0,1] as subspaces of R with the usual topology are not homeomorphic.

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Q: The vector space C[a, b] of all continuous real-valued functions on [a, b] is not the span of any…

A: A vector space is a set whose elements, often called vectors, may be added together and multiplied…

Q: (b) Prove that if V is finite dimensional, then every subspace of V is also finite dimensional.…

A: (b)