Let B and C be the subsets of some universal set X then prove that (C N B')' = BU C'

Q: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

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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: Prove that the set (0. 1) = { all xE RIO <x< 1} is uncountable/non-denumerable.

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Q: Let (X,T) be a toplogical space, A, BC X. Show that АС В Ас в, АсВ. 介

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Q: Let (R,r) be topological space with usual topology on R, then the set Qof rationals is O Neither…

A: ℚ is the set of rational numbers. ℝ is the set of real numbers. So, ℚ is a subset of ℝ .        i.e.…

Q: Prove that there is no set to which every se Show that for any nonempty set A, there xE Bex belongs…

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

Q: [3.12] Let S be a nonempty convex set in R", and let f. R" → R be defined as follows:

A: Let S be a nonempty convex set in Rn, and  Let f:Rn→R be defined as follows:…

Q: X, Y and Z be subsets of the set . Verify the following propert XU(YUZ) = (XUY)UZ, XN(YnZ) = (XnY)…

A: X, Y, Z be subset of a universal set Ω to verify…

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: Let S and T be sets. Prove that SUT = T iff S is a subset of T.

A: To Prove S∪T=T iff S is a subset of T

Q: (2) Show that if f has support in a compact set A CR and g has support in a compact set B C R then f…

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Q: Let (X, E, µ) be a measure space. Justify through an example that the hypothesis µ(X) < o in…

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Q: Show that if A is closed in Y and Y is closed in X, then A is closed in X.

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Q: Consider two sets, A = {1, 2, 3, ..., 100} and B = {3, 5, 7, 9, ... , 201}. Prove that |A| = |B| by…

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Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

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Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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

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Q: Give an example of a compact set A and a closed set B in R2 such that (conv A) N (conv B) = Ø but A…

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Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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Q: Let X be a topological space. Prove that (a) Ø and X are closed. (b) Intersections of any number of…

A: Union of any open sets is open and intersection of two open sets is open.

Q: Consider the non empty sets A and B such that A and B have the same cardinality. Prove, for any set…

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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: Prove that if S is a bounded set then S is bounded.

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Q: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, is a topology on X.

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

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: 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: Give an example of a closed subset S of R^2 such that conv S is not closed

A: An example of a closed subset S of R2 such that conv S is not closed is given below.

Q: Prove that a set A is countable if and only if there exist a surjection from B onto A.

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Q: 7. Let X be a set with |X| = n € Z4. Prove that for any set Y, |Y| = n if and only if there exists a…

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Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

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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 and Y be two discrete spaces, then* O x is never homeomorphic to Y O X is homomorphic to Y if…

A: As two discrete spaces are homeomorphic iff they have same cardinality .

Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

Q: Prove that if R, S, and T are nonempty sets, then R×T ⊆ S×T if and only if R ⊆ S.

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Q: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

Q: 2. Let X be any uncountable set, Prove that: 7. = (GC X: G is countable} U {0} is a topology on X.

A: Since you have posted multiple question... according to company rule we are supposed to answer first…

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

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Q: 3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a)…

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

A: Here is the proof .

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 with more than one element. (X,7) is a topological space such that Vxe X, {x} €T then…

A: Given the set X contains more than one element.

Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

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Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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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: 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: 2- In a metric space (X, d), A is the intersection of all closed subsets of X which contains A.

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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 every set with positive Lebesgue measure contain a non-measurable set

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

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.