Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7. (i) Prove that |X ∩ Y| ≥ 4. (ii) Deduce that X ∩ Y ∩ Z is non-empty. [Hint: Consider (X ∩ Y) ∪ Z.]

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

A: Let (S, d) be a metric space. Consider a closed ball Srx0 with both center x0∈S and radius r>0…

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

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Q: 2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p…

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Q: 2. Let S be a bounded set in R and. let a < 0. and let aS := {as : s E S). Prove that inf(as) = a…

A: Let S be a bounded set in R and let a&lt;0. And let aS := {as : s belongs to S}. We have to prove…

Q: Prove that the set S = {n² : n e N} is denumerable. %3D

A: We have to show that the given set S is a denumerable set.

Q: Every closed and bounded subset of X is compa The subset {z EX: |||| ≤ 1} of X is compact... X is…

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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: Theorem 6.2. Let C be a compact subset of Rstd· Then C has a maximum point, that is, there is a…

A: We start the proof by assuming there are no such maximum element in C.

Q: 4. Assuming that the c possible subset of R (c) f(x) = V5 –

A: Use the definition of domain and range to solve this problem

Q: F, there is a disconnection A, B of F such that x E A and y E B. 8.X. If C1, C2 are connected…

A: We use the fact that a set X is connected if there is no continuous function form f from X to Y=…

Q: In a Cofinite topology of X the collection of subsets a) r = {AcX|A ±Ø or A is finite} b) r = {Ac X|…

A: Let X be a non-empty set, then the collection of subsets of X whose complements are finite along…

Q: Give an example of a set X and topologies T1 and T2 on X such that T1 union T2 is not a topology on…

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Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Two discrete spaces are homeomorphic if there exists a bijection between them and it is possible if…

Q: If S, T are nonempty bounded subsets of R with S ⊆ T, then inf T ≤ inf S ≤ sup S ≤ sup T.

A: Since S ⊆ T it follows taht inf T ≤ t, ∀ t ∈ T, then  inf T ≤s , ∀ s ∈S:Therefore inf T is  a lower…

Q: 2. Let S is a subset of R. Prove that S is closed if and only if S contains all of its accumula-…

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Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

A:

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

A: option (c) is correct.

Q: Prove that if ø is convex, then a-[$(x + Az) – Þ(x)] is nondecreasing for å > 0.

A: As  φ is convex we conclude that its second derivative :  φ'' is always positive ∴  φ''(x) &gt; 0…

Q: If x is a lower bound for S in R then x must be unique..

A: If x is a lower bound for S in R then x must be unique.

Q: Show that there exists a set N C (0, 1) that is not Lebesgue measurable.

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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: Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set…

A: Given that A is a subset of the topological space X and p is a point in X. Consider a sequence…

Q: 2. Show that the set 3 S = 3 2 2 2 2 spans all of R³ or find an element of R' not belonging to…

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Q: 9. Let be a bounded open subset of C, and 4: Prove that if there exists a point zo E such that 2 a…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Suppose that S is a nonempty subset of R and k is an upper bound of S. The number k = sup S if and…

A: Given: S is a non-empty subset of R and k is an upper bound of S. To prove : Number k=sup S if and…

Q: 2 49) Given nonempty subsets of R², say Y....., let Y = nin enner-xer Fix P E R². For a nonempty…

A: Given: Non empty subset To find: Solution

Q: Q3. Let N = {1, 2, 3, ...} equipped with the cofinite topology. Prove that M = {7,14, 21} is a…

A: Given below the detailed solution

Q: Let E be a non-empty bounded subset of R. Prove that inf E is unique

A: Given: E is a non-empty bounded subset of ℝ. To show: inf E is unique.

Q: (c) Prove that only Ø and R are open subsets of R

A: A subset Uof a metric space (M, d) is called open if, for any point x in U, there exists a real…

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

A:

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

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Q: Let x, y ∈ Z. Prove that x − y is even if and only x and y are of the same parity.

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Q: 1. Let T and S be topologies in X. Show that SnT is also a topology in X.

A: 1.

Q: 9.C. Prove directly that if K is compact in Rº and F is a closed subset of K, then F is compact in…

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Q: Theorem 6.17. Every compact subset C of R contains a maximum in the set C, i.e., there is an m E C…

A: To prove that, every compact subset C of ℝ contains a maximum in the set C, that is, there is an m∈C…

Q: 3. Let E be a nonempty subset of R that is bounded above, and set U = {ß ER is an upper bound of E}.…

A: Given that Let E be a nonempty subset of R that is bounded above, and set U = { B belongs to R is an…

Q: Prove that a set A is countable if and only if there exist an injection of A into N.

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Q: Let (X. T) be a T, topological space and A a nonempty subset of X, if x is a cluster point of A…

A: Given: Let (X,T) be a T1 topological space and A a non-empty subset of X, if x is a cluster point of…

Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

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Q: 12. If E is a bounded above subset of Z, then A. sup E exists and sup E E E B. sup E exists and sup…

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Q: 11 Prove that if A CR and |A| > 0, then there exists a subset of A that is not Lebesgue measurable.

A: As per the policy, we are solving a first question. Please repost the question and specify which…

Q: Give an example of a collection of compact subsets in R, say (Kn) n=1 to infinity,  such that U n=1…

A: Give an example of a collection of compact subsets in R, say (Kn) n=1 to infinity,  such that U n=1…

Q: Let A be dense subset of (X, t)

A: Given: Let A be dense subset of x,Τ and let B be non empty open subset of X. To find: Fill in the…

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

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Q: 1/ For any. topological space (X, T), if T has only three elements then (X, T) is connected.

A: Given: Topological space X, T and T has 3 elements  To prove that X, T is connected when T has 3…

Q: Find an infinite collection {Kn: n N} of compact sets in R such that the union U K₂ is not compact.…

A: We know the definition of compact sets , A subset S of ℝ is said to be compact if every open cover…

Q: Let E1, E2 be two compact subsets of RP. Use the Heine-Borel Theorem to prove that EU É, is also a…

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Q: Q Let X be an Infinite sel with topdogy ?. Prove or disprove : (a) If every mfinite Suhsel of x i…

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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.