Suppose a set M is closed and bounded. Prove that M has a maximal and minimal element.

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

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Q: Suppose that A and B are subsets of R which are nonempty and bounded above, and suppose that BCA.…

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Q: {1, x + x², x²} ≤ P2. Show that this is a spanning set for P2

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Q: Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

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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: Give an example of a set w-limit that is a minimal set.

A: To give an example of a set w-limit that is a minimal set.  Given a collection of sets , a member…

Q: Q1\ Define the compact set. Then, prove that: if A is a compact subset of F , there exists a best…

A: Compact set : A set is compact if and only if every open cover has a finite subcover. There is also…

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

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Q: A and B are bounded subsets of R, then prove that UB and AnB are bounded.

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Q: Prove that an infinite subset of a compact space has a limit point.

A: We can Prove an infinite subset of a compact space has a limit point.

Q: Show

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

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: (a) Suppose S is a subset of positive integers, ZZ. If |S| ≥ 3, prove there exist distinct z and y…

A: (a) Given that S is subset of  such that Observed that any positive integer is either even or odd…

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

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Q: Exercise 4. A subset A of a topological space X is called dense if cl A = X. The set Q of rational…

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Q: Show that the only Hausdorff topology on a finite set is the discrete topol- ogy.

A: The question asks us to prove that the only possible Hausdorff topology on a finite set is the…

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: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

Q: Let A = {2, 3, 4, 6, 10, 12, 16} and R is a partial order relation defined on A where aRb if and…

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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: Let R be the set of all real numbers with the euclidean topology. Show that R has an uncountable…

A: Let R be the set of all real numbers with the euclidean topology. Show that Rhas an uncountable…

Q: The connected components of a space X are all the maximal connected subsets with respect to…

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Q: a)Let A = {x ∈ R + : x 2 < 2} . Explain why A i bounded above and below and find sup(A) and inf(A).…

A: A can be written in interval form

Q: The intersection of any countable family of

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Q: Let A = {2, 3, 4, . . .} be ordered by “x divides y”. (i.) Determine the minimal elements of A.…

A: Let  be ordered by “ divides ”.(i). Determine the minimal elements of .(ii). Determine the maximal…

Q: X is an infinite well-ordered set without maximum element. Show that any element of X has an…

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Q: 1) Let Xbe an infinite set with the Finite Complement Topology. Let A EX be an infinite subset.…

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

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

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Q: 5. Suppose that A and B are bounded subsets of R. Prove that AUB is bounded and that sup(AUB) =…

A: Given that A and B are bounded subsets of R.   We have to Prove that AUB is bounded and that…

Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

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Q: If A is bounded from the below, then the set of all lower bound of A has infimum but may or may not…

A: Given:               if A is bounded from the below, then the set of all lower bound of A has:…

Q: Let A = {2, 3, 4, 6, 10, 12, 16} and R is a partial order relation defined on A where aRb if and…

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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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