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

Q: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

A: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

Q: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

Q: (2) Find all connected covering spaces of RP2 v RP².

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Q: Find an open cover of the interval (−∞, 3]. Justify why that open cover has no finite subcover.

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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, 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: 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: Prove the following theorem: If ƒ(x, y) is defined in an open region R of the xy-plane and if ƒx and…

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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: Prove that a square maximizes the area of all rectangleswith perimeter P.

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Q: Give the set of limit points A0 of a singleton A = {(5, 2)} on the plane R 2 with the discrete…

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

A: Answer is mentioned below

Q: If X is inner product space over real numbers then the inner product always greater than or equal…

A: The objective is to determine whether the given statement is true or false. Given statement is, " If…

Q: Each nonempty can be represented as the convex hull of its extreme points. --------

A: Each of the nonempty compact convex set can be represented as the convex hull of its extreme point.

Q: Show that the maximum norm on C[a, b] is not induced by an inner product

A: to show that the maximum norm on Ca,b is not induced by an inner product . as we know , a norm X,·…

Q: (K", || |,) is called the LP-norm on K". is a normed linear space for 1 <PS00. || |,

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Q: What is the norm of a partition of a closed interval?

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: If X is inner product space over real numbers then the inner product always greater than or equal…

A: FALSE .  (by the definition)

Q: Show that in R with the usual topology, the only open sets connected by trajectories are the open…

A: Given that R with the usual topology  A collection of of subsets of R which can be expressed as a…

Q: 3) Consider the topology ↑ {GCR: 0 ¢ G or G = R} on R. Is open = %3D interval (-1, 1) compact in (R,…

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Q: Show that close ball Y in a metfic space (X,d) is a closed set also show that if (X,d) is complete…

A: Show that closed ball Y in a metric space (X,d) is a closed set also show that if (X,d) is complete…

Q: Show that the inner product (,) on an inner product space X is a bounded sesquilinear form h. What…

A: Given that X is an inner product space with an inner product ·,· defined on it. We know that a map…

Q: Letxandy and z be normed space s and s: xy andTIYz bounded oper atorsprove

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Q: Prove the following sets are not compact by finding an open cover that does not have a finite…

A: a) Given-  A set A=−1,1. Explanation- Consider a set Xn=-2,1-1n for n=1,2.... Then clearly,…

Q: Show that the compactness in satability theorem is neccessary.

A: The Stability Theorem fails when X is not compact

Q: umat is bounded, and is not compact in Q.

A: Here the closed means closed with respect to subspace topology. And compact means every cover has…

Q: Without appealing to Heine-Borel, prove that [0, 1] CR is compact and connected in the standard…

A: Please check in the next step

Q: Describe the largest set S on which it is correct to say that fis continuous. AX, y) = In(1 + x² +…

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Q: Show that the maximum value of a2b2c2 on a sphere of radius r centered at the origin of a Cartesian…

A: Let fa,b,c=a2b2c2 subject to a2+b2+c2 =r2  The gradient equation is…

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

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Q: Show that every closed ball in R" is an n-dimensional manifold with Assuming the theorem on the…

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

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Q: Can you show an example of Null space?

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Q: Show that the closed ball Y in a metric space (x,d) is a closed set .Also show that if (x,d) is…

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Q: Find the dimensions a and b of the rectangle of the maximal possible area inscribed in a semicircle…

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Q: (a) Prove that every nonempty proper subset of a connected space has a nonempty boundary. Is the…

A: As per Bartleby guidelines, for more than one question asked, only first is to be answered. Please…

Q: Define what it means for a topological space to be completely regular,

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Q: topological spaces X and Y,

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Q: Consider the following subsets of real plane R termine with graph: Arcwise connected A = ((x,y): 0 ≤…

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Q: Show that the compactness in satability theorem is neccessary

A: Stability Theorem :- The following classes of smooth maps on a compact manifold X in to Y are stable…

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

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Q: Let (X, τ ) be a discrete space. Is (X, τ ) second countable? Explain why

A: Consider the provided question, Let x,τ be a discrete space,    ⇒x,τ  be denote topology if x,τ is…

Q: Is any discrete space complete? Give an example to substantiate your answer.

A: A metric space is complete if and only if every Cauchy sequence is convergent.

Q: So F. dr - 0 for every closed curve Cin R, an open connected region. True False

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Q: the euclidean space R³ is a separable metric space. true or false?

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…

Q: Construct an open cover for the interval [0, 1]. Verify that the open cover is in fact an open…

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

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Q: Qz\ show that ?_C=0} is not a swbopace of R

A: