For each condition below, give an example of a set that satisfies that condition, or prove that one does not exits (a) An open set that is a subset of Q (b) A non-empty open set that is a subset of Q (c) A non-empty closed set that is a subset of Q (d) Two disjoints open sets whose union is R (e) Two non-empty disjoints open sets whose union is R (f) An infinite subset of R with no limit points. (g) A bounded infinite subset of R with no limit points. (h) An infinite union of closed sets that is not closed.

Q: i. PROVE THAT THE EXCLUDED POINT TOPOLOGY ON ANY SET WITH AT LEAST 2 POINTS IS NOT HAUSDORFF. ii.…

A: Definition: A topological space X, τ is said to be Hausdorff if for every x, y∈X there exits U, V∈τ…

Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: Let X={a,b,c} and Y=∅. What is X∩Y? If the set is empty type ∅, otherwise write your answer in…

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Q: Assume that the universal set is Z. Consider the following sentence: (3t = Z) (t •x = 20). (a)…

A: Here, in the question, it says, such that tx= 20.We know a statement in logic and mathematics is a…

Q: Let & = (A xEA) be a family of sets, and let B be a set, Then B- (UA) = U (B-A). «ΕΔ a EA True False

A:

Q: display that if A is an infinite set, then whenever B is a set, A ∪ B is also an infinite set.

A:

Q: (1) If A is an uncountable (2) Z† × Z is a countable set. Select one of the following choices: set…

A: A countable set is a set that can be put into one-to-one correspondence with the natural numbers…

Q: Determine whether the following statements are true or false: (a) The empty set ∅ is a subset of {1,…

A: Empty set is a subset of every set. A Power Set is a set of all the subsets of a set.

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

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Q: If G(sub alpha) is a collection of closed sets, then the intersections of the G(sub alphas) is a…

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

A: A non empty set S is called bounded if it has both upper and lower bounds, In a certain sense a non…

Q: Let A be a partially ordered set that has the least upper bound property. Then every nonempty subset…

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Q: Write out in words the description of the set. {x|x  , x ≤ 10} The set of        , x, such that x…

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Q: open

A:

Q: Provide examples of sets A and B where strict inequality sup(A+B) < sup A+ sup B holds.

A: We have to given an example to prove given inequality:

Q: Prove that a finite union and an arbitrary intersection of compact sets is a compact set.

A: Step 1: Step 2: Step 3: Step 4:

Q: Construct a family of sets that demonstrates R is not compact

A: