8. (a) Show that the intersection of finitely many open sets is open.

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Q: 3.1 Show that a finite intersection of closed sets is closed.

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A: To show that compact sets are closed under arbitrary intersections and finite unions

Q: Show

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

Q: 1. Prove: N × N is countable. 2. Prove: The Cartesian product of countable sets is countable. 3.…

A: Since you have posted multiple questions. But according to guideline, we will solve first question…

Q: 1. Find all regular points of the set : {x € Rª | 2(x1+x2+x3)³ +3(x1+x2+x3)² = 1, x²+x²+x²+x4 = 1}.

A: The given set is: x∈R4|2(x1+x2+x3)3+3(x1+x2+x3)3=1,x12+x22+x32+x4=1

Q: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

Q: Let ∅ be the empty set. Let X be the real line, the entire plane, or, in the three-dimensional case,…

A: Given:

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

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Q: 1. a) Does every bounded infinite set of real numbers have a limit point? Either show that it does,…

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Q: Use the method of intersection of sets to compute the following. (a) GCD(12,16,8) and LCM(12,16,8)…

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Q: Prove that the cartesian product of a countable set is countable

A: This is question related to set theory's topic countability.

Q: Prove that the unoin of a finite number of bounded sets is bounded.

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Q: Prove that the intersection of any two open sets is open

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Q: 1) Let f: A ->B with C1 C C2C A and Dic D,CB Which is true? a) f(C) c f(C2) OR b) f(C2) c f(C:) (i)…

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Q: Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

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Q: Let A = {armadillo, bear, crocodile, dolphin, eagle, fox}. Construct a par- tial ordering < on A…

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Q: {x, x + 1, x2 + x + 2, x2 + x + 1} a spanning set for P3?

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Q: Prove the empty set hasn't supremum and infimum

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Q: Let N (A) A = = span 4 5 -6 -2 -2 6 2 -4 - 10 –7 6 2 Find a spanning set for the null space of A. -4…

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Q: 8. Let A, B and C be sets. (a) Suppose that AC B and BC C. Does this mean that A S C? Prove your…

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Q: 4. Let (X,≤) be a partial ordered set. Show that a subset A of X has at most one least upper bound.

A: Given that   be a partial ordered set and A is a sub set of X.Now we want to prove that A has at…