Exhibit a countable collection of open sets U; such that each open set OCR can be written as a union of some of the sets U;.

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

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Q: (a) The intersection of any collection of closed sets in X is closed (in X). (b) The union of any…

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Q: show that the Cantor set is a closed subset of R and has measure zero.

A: We know the definition of Cantor set, The Cantor set is defined as the intersection C = ∩n=1∞Cn ,…

Q: A metric space (X , d) and a collection of open sets U such that UEU is not open.

A: Let X, d be a metric space. Let U be the collection of open sets in the metric space X, d. That is,…

Q: Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: How do I write a set T such that S is an element of T and a proper subset of T?

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

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Q: Show

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

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U ·. U Bn equal A exactly?…

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Q: Suppose a set A is uncountably infinite, that U is countably infinite, and that UC A. Show that A -…

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

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Q: Give an Such that. example of sets A andB

A: Universal Set: We consider all the sets to be the subsets of a given set known as the universal set…

Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: Prove that for all sets A and B, A – B = AN B

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

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Q: 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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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: Let A and B be subsets of a universal set U, Prove that ACB if and only if B C A.

A: By definition, B′⊂A′ implies exactly two things: Every element of B’ is in A’ There exists an…

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U... U Bn equal A exactly?…

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Q: 3.6 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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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: Prove that if A is a set then so is {A}!

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

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

Q: Prove that the Cantor set has measure zero.

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Q: Is there a set A that satisfies A = {A}? If yes, exhibit one such. If not, Why not ?

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Q: Prove that every infinite set S contains a proper subset similar to S.

A: We have to prove that every infinite set S contains a proper subset similar to S. Let X=S Let X be…

Q: Use the definition of set operations to prove that for any set A, A- A = Ø . A + A = Ø

A: To prove: Using definition of set operations, for any set A prove that                  a) A-A=ϕ…

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: Is there a set A that satisfies A = {A}? If yes, exhibit one such. If not, Why not exactly

A: Set equality

Q: Show that the derived set E′ of any set E is closed.

A: Here is the proof .

Q: The closare of a set is smallest closed set containing s صواب ihi

A: Let S be a subset of X: S⊆X. Let M be the closure of S. Then M is the smallest closed set which…

Q: open

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Q: The intersection of any countable family of open dense-sets in a locally compact space X is dense.

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Q: Construct a family of sets that demonstrates R is not compact

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Q: Let A be a well-ordered set, then every subset of A is not well-ordered

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Q: 1. Prove that a set SCR is everywhere dense if and only if S intersects every non-empty open set in…

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Q: Every proper simply connected oper equivalent to an open subset of the

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Q: Let A be a set. Show that ∅×A = A×∅ = ∅.

A: A×B=A×BIf A=∅ or  B=∅,thenA×B=A×B=0Since a set with 0 cardinality must be an empty set, we have…

Q: open ball is the open set

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

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Q: Prove that a set A is connected iff A is and interval.

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Q: B. Prove that the union of a finite number of compact subsets F1, F2, ..., F, of RP is a compact…

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

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Q: 2. Prove that the union of any finite collection of closed sets in RP is a closed set in RP.

A: A set A is closed if and only if every limit point of A belongs to A. Now we can prove the result.…

Q: Let A1, · · · , An be countably infinite sets. Prove that A1 × A2 × · · · × An is countably…

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Q: Consider the set X={a,b,c,d} construct all the possible topologies of X that have only three open…

A: Given: The set X=a,b,c,d To find:  all the possible topologies of X that has only three open…