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

Q: Let A be a set. Suppose that f: w→ A is onto A. Prove that A is countable.

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Q: If A and B are bounded subsets of R, then prove that AUB and AB are bounded.

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Q: 5. Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E) %3D

A: This is a problem of Real topology, metric topology.

Q: Let S be a nonempty subset in R". Prove or disprove by a counterexample that Span S = (S+)+

A: We have to prove or disprove by counterexample that Span S=S⊥⊥

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A: The "finite closed topology" describes the closed sets. A subset U is open if and only if U=∅ or X∖U…

Q: Suppose that A and B are bounded, nonempty subsets of R. Prove that AU B is bounded and that sup(AU…

A: Consider the provided question, Suppose A and B are bounded, nonempty subsets of R.

Q: If S is compact and D is in S is a closed set, then D is compact. Use the definition of…

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Q: Use Induction to prove for any non-empty set A and any partition of the set B by non-empty sets…

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Q: Let A be a compact set in R. Show that the complement A is open and contains an interval of the form…

A: Solution: Before going into the problem we must know the Theoem named Heine-Borel Theorem.…

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: For any collection a of subsets of X, there i ntaining a.

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Q: ] Prove that a nonempty closed subset of R, if it is bounded from below, has a least element.

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Q: Let X be an infinite set. (a) Show that is a topology on X, called the finite co со

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Q: 5. Show that if A and B are bounded subsets of R, then A U B is a bounded set. Show that sup(A U B)…

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

A: In this question, we use the definition of compactness to prove that a closed subset Y of a compact…

Q: Prove that if A is a nonempty finite set and B is denumerable, then A × B is denumerable.

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Q: Prove that every denumerable set A can be partitioned into a denumerable number of denumerable…

A: In many books you see different kinds of proof , but I can explain you it easily by "pigeonhole…

Q: Exhibit a countable collection of open sets U; such that each open set OCR can be written as a union…

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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: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

A: Let A be a non-empty compact subset of ℝ Let B be a non-empty closed subset of ℝ such that A∩B=∅ To…

Q: Every connected subset of R is complete is ture or false?

A: Every connected subset of R is complete is true or false?

Q: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

Q: The Cantor set is closed .A O dense .B O open .C O neither open nor closed.D O

A: Closed

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

A: "Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact." The…

Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

Q: Prove that the Cantor set has measure zero.

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Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

Q: Specify the set A by listing its elements, where A={x:x is a whole umber less than 100 and divisible…

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Q: Prove that every finite set S of R" has no accumulation point.

A: Now taking set S Let ? = {?1, ?2, . . . , ?n }.

Q: If A and B are bounded subsets of R, then AuRand AOR are bounded: prove that

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Q: Let A, BCR be two non-empty bounded sets. Assume that for every x E A, there exists y E B such that…

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Q: compact, then E must be infinite. Exercise 6.46. Prove that if E is a closed subset of a compact set…

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Q: Set A is bounded below if and only if set -A is bounded above. True False

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Q: Give an example of a closed subset S of R2 such that conv S is not closed.

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Q: Let F be an event space. Show that F is closed under countable intersections.

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Q: Assume that A is a σ-algbera of subsets of ℝ. Prove: If A contains all intervals of the form (a,∞),…

A: A σ-algebra on a set X is a collection of subsets of X that is closed under complement and countable…

Q: Let $X$ be a set and assume that p E X. Let T, be the collection (Ø, X, and all subsets of X that…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: show by an example that a nonempty proper subset of S in the metric space (S, d) can be both open…

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Q: Find an example of a nonempty set satisfying: *Not Open, Not Closed, Dense in R *Open, nowhere…

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Q: Determine if the two sets are dense in R, nowhere dense in R or somewhere in between: A=Q⋂[0,5]…

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

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Q: Prove that a set S⊂R is everywhere dense iff it intersects every non-empty open sets in R. Prove…

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Q: Let $X$ be a set and assume that p E X. Let Tp be the collection 0, X, and all subsets of X that…

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Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤…

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Q: Let A, B be disjoint infinite sets. Prove that their union A U B is denumerable if and only if both…

A: Given A , B be disjoint infinite sets. Firstly, consider A∪B is denumerable Since, A⊆A∪BandB⊆A∪B…

Q: Prove that if A and B are disjoint denumerable sets, then A UB is denumerable.

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Q: Suppose A is a nonempty subset of R. (a) Is it true that Là=L? Prove or disprove. (b) Let iso(A)…

A: Given that A is a non-empty subset of ℝ. (a) To prove that                      LA¯ = LA

Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup up A exist? Show that (a) inf…

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Q: Let A be a nonempty subset of R that is bounded below. Define the set B={b: b is a lower bound of…

A: Given A is nonempty subset of ℝ. Also, A is bounded below. Greatest lower bound property: The…