Which of the following is true: Every bounded subset of (R, Tusuat) is compact. Every closed subset of (R, Tusual ) is compact. Every compact subset of T2-Space is clopen. Every closed subset of (R, TCOF ) is compact.

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Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

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Q: Show that every countable subset of R" has Hausdorff dimension 0.

A: claim- to show that every countable subset of ℝn has Hausdorff dimension 0. it is known that…

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Q: X, Y and Z be subsets of the set . Verify the following propert XU(YUZ) = (XUY)UZ, XN(YnZ) = (XnY)…

A: X, Y, Z be subset of a universal set Ω to verify…

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Q: Let X be a locally compact Hausdorf f space

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A: The given statement that " A normed space X over R is seperable if it has a countable dense subset "…

Q: 25. For any set S CR, let S denote the union of all the open sets contained in S. (a) Prove that S…

A: The given question is solved below.

Q: Exercise 3. In this exercise (X, d) is a metric space. .} 1) Assume that X is compact. For every r>…

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Q: Exercise 1. Give examples of subsets A, B in a topological space X (for example X = R or X = R²)…

A: To give an example of subsets A,B in a topological space X such that both A,B are connected but A∪B…

Q: Use the definition of compactness to prove that a closed subset Y of a compact set X ⊆ R is compact

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Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

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Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

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Q: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

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Q: Let (Ka)aeA be a collection of compact subsets of a metric space, X. Show naeAKa is closed in X.

A: A metric space X is compact if every open cover of X has a finite subcover.

Q: Show that every normed liner space which is linearly isometric with a separable space is separable.

A: A normed linear space or normed vector space is a vector space over the real or complex numbers,…

Q: 12 Let (RTR) is topological Space Such that TR-243 U2GSR=R-G-G is finste} Find buz), Z², Z. z' then…

A: Here given ℝ, Tℝ is a topological space such that Tℝ=ϕ∪G⊂ℝ: ℝ-G=Gc is finite This topology is known…

Q: Suppose X is a connected topological space with the property that every point x of X has a…

A: In this problem, we consider a connected topological space X with the property that every point in X…

Q: Prove that a closed convex set in R" has a finite number of extreme points if and only it is the…

A: A closed convex set in ℝn has a finite number of extreme points if and only if it is the…

Q: Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact. * O…

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Q: Use the definition of compact set to prove that the set [0,1] is compact

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Q: Exercise 10: Show that if A and B are bounded subsets • bounded set. Show that then AUB is a of IR,…

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Q: ## Commpact subse regular Space X and if of is subset of A such that .cey th there is a closed a…

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Q: Consider the subsets [0, 1] and [3, 4] of R, the set of all real numbers with the euclidean…

A: Consider the subsets [0, 1] and [3, 4] of R, the set of all real numbers with theeuclidean topology.…

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: Show that an open subcover R is not compact by finding cover that does not have finite

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Q: determine an b) Every Space CD Every non True or False, Justify why, not totally bounded Open Set S…

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Q: Let (X.J) be a topolgical space and {C1, C2 Cn} be a finite collection of compact subsets of X.…

A: Given Let (X,I) be a topological space and C1,C2,......Cn be a finite collection of compact subset…

Q: Let (M, d) be a metric space and K ⊂ M. The following statements are equivalent: (i) K is relatively…

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

Q: If (X,7,) and (X,7,) T. - spaces, then (X,7,nt) is also a T,- space. GOOD LUCK

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Q: Let x be non-empty of points. a topological Space having open set with finite numb cal shase wh

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Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

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Q: Exercise 1. Give examples of subsets A, B in a topological space X (for example X = R or X = R²)…

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: A property is said to be topological property if it is preserved by homeomorphism.

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Q: a Normed &pace and and CL(X,Y) denste, of bounded Let X,Y be a Suppose BL(X,Y) respectively, the…

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