open ball is the open set

Q: Prove that a subset of a set of measure zero has measure zero.

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Q: Show in the discrete metric space: 1. All sets are open (especially single points);

A: Let X be a non-empty set and d be discrete metric on X

Q: If  X is a countable set show that every topology on X is second countable.

A: To Show: If  X is a countable set show that every topology on X is second countable.

Q: In the discrete topology on the set X, every set is open and it also follows that every set is…

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Q: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

Q: 6. Show that the span of this set is all of R?. {}}

A: Let S=v1,v2,...,vn be a set of vectors in a vector space V(F) . The vector space V is spanned by S,…

Q: Show that the sets [0,1] and (0, 1) are equinumerous 7.

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Q: Prove that the smallest member of a set, if exits, is the greatest lower bound of the set.

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Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: Show that an infinite intersection of closed sets F, k = 1, 2, 3, . .., in a metric space (S, d) is…

A: first we will proof that an infinite union of Open sets in metric space ( S, d) is open using the…

Q: that the boundary of a closed set is nowhere dense in

A: Matric space

Q: let x be an infinite set and let ta be a topology on x in which all infinite subsets of x are open…

A: Given that X be an infinite set and let τ be a topology on X in which all infinite subsets of X are…

Q: Prove that intersection of any class of convex subsets of N is convex?

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Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

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Q: Prove that ifS is a bounded set, then inf(S) = -sup(-S) %3D

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Q: Show

A: We prove this statement by using definition of Borel Set.

Q: Let A be a bounded non-empty set. Then inf A < sup A. True False

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Q: Let A be a denumer able set. Prove that A has a denumer able subset B such that A - B is…

A: Let A =set of multiples of 4, B=set of multiples of 3. Both are denumerable. Intersection of A and B…

Q: Use the Heine-Borel theorem to give an example of a set not compact in R with the usual topology…

A: An example of a set not compact in R with the usual topology and a non-compact set R2 with the usual…

Q: Show that every open set with respect to 7 is a union of open ball(s).

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Q: Prove that empty set is subset of all sets: φ ⊆ A

A: Given: An empty set and set A is given. To Find: To prove that empty set is subset of all sets: ϕ⊆A

Q: If X is a normed space then the open ball and closed ball is a Example: convex set

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Q: Show that the o-algebra B of Borel sets contains all F,-sets.

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Q: 4. Let A be an open set in å nörmed space subset B of X .

A: Que  Ans :  Given ( X, ||-||) be a norm linear space,  A be any open subset of X    ( def- set A is…

Q: Show that any convex open set is simply connected.

A: The objective is to prove that any convex open set is simply connected. Simply connected is nothing…

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

A: We know that a set E is compact if every open cover of E has a finite subcover. We suppose, that O…

Q: Prove that every Union of Open sets is also Open.

A: To prove: Every Union of Open sets is also Open.

Q: Suppose S and T are balls with the same center. Show that one of them must be a subset of the other.

A: It is given that S and T are balls with same center. Let the center of the balls be a. Assume that…

Q: union of a collectionm of connected sets that have a point in common is conne

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Q: Prove that the unoin of a finite number of bounded sets is bounded.

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Q: Prove that the greatest member of a set, if exits, is the least upper bound of the set.

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Q: Let X be an infinite set, then any topology on X is also infinite. Select one: O True O False

A: Note: " Since you have asked multiple question. As per our guidelines we are supposed to solve only…

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

A: Topology Question

Q: prove that empty set is a subset of every set

A: We have to prove that empty set is a subset of every set = ?

Q: Prove that a set is closed if and only if its complement is open

A: prove of the given problem is given below...

Q: a. Show that the null-space N(A) is a convex set. Is it true for every A?

A: As both questions are of different type,we shall solve first Que only. For other questions kindly…

Q: Prove that every nonempty subset of a well-ordered set is well-ordered.

A: Given problem is :

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

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

Q: Show that any subset of bounded set is bounded

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Q: open

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Q: Show that closed ball in metric space (x,d) is a closed set. Also,

A: Given: (x,d) is a metric space The objective is to show that the closed ball y in (x,d) is a closed…

Q: Prove using Venn diagram that An (B U C) = (A n B) U (A n C)

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Q: Prove that the open ball B(p, 8) = {x : ||x – p|| < 8} is a convex set. [Hint: Use the Triangle…

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Q: In any metric space, the only open and closed set at any time is the empty set and X only True False

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Q: Prove that every set with positive Lebesgue measure contain a non-measurable set

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Q: Show that both ℝ and ℝ2 may be covered by countably many open balls

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Q: )Š is the smallest closed set in M such that S CS.

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Q: Prove that any ball (open or closed or half open) in R³ is path connected.

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Q: Prove that the union of any number of open sets in Ris open.

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