Real Analysis MSQs and MSQs : Point Set Topology - III

25. Select the incorrect statement about the set \(A=\left\{1,1+\frac{1}{3},1+\frac{1}{3}+\frac{1}{3^{2}},\ldots,1+\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}\ldots+\frac{1}{3^{n-1}}\right\}\).

(a.) The set \(A\) is bounded.
(b.) The inf \(A_A\) is 1.
(c.) The set \(A\) has its largest elements.
(d.) None of these.

26. Let \(S=\left[(-1)^n\left(\frac{1}{4}-\frac{4}{n}\right): n \in \mathbb{N}\right]\). What are \(\sup S\) and \(\inf S\) respectively?

(a.) \(\sup S=\frac{1}{4}\) and \(\inf S=\frac{1}{4}\)
(b.) \(\sup S=\frac{15}{4}\) and \(\inf S=\frac{-7}{4}\)
(c.) \(\sup S=\frac{15}{4}\) and \(\inf S=-\frac{1}{4}\)
(d.) None of these

27. Let \(S\) be a subset of \(\mathbb{R}\) such that \(\inf S=\sup S\). What can be said about \(S\)?

(a.) \(S\) is empty
(b.) \(S\) is a singleton
(c.) \(S\) is finite but may not be a singleton
(d.) Can't say

28. Let \(S=\left\{\left(1-\frac{1}{n}\right)\sin\frac{n\pi}{2}: n \in \mathbb{N}\right\}\). Which statement about \(S\) is correct?

(a.) \(S\) has the smallest element but not the greatest element
(b.) \(S\) has the greatest element but not the smallest element
(c.) \(S\) has both the greatest and least elements
(d.) \(S\) is bounded but does not have both the smallest and greatest elements

29. Which of the following sets is not an \(nbd\) (neighborhood) of each of its points?

(a.) Set \(\mathbb{Q}\) of rational numbers
(b.) Set \(\mathbb{Q}^c\) of irrational numbers
(c.) Set \(\mathbb{Z}\) of integers
(d.) None of these

30. Let \(S\) be an uncountable set and \(T\) be a set of those real numbers \(x\) such that \((x-\delta, x+\delta) \cap S\) is uncountable. Which of the following statements is/are correct?

(a.) \(T\) is countable
(b.) \(S-T\) is countable
(c.) \(S \cap T\) is uncountable
(d.) All of the above are correct

31. Let \(E_{i}\) be subsets of \(\mathbb{R}\) such that \(E_{2} \cap E_{i}^{\prime}=\phi\). Then select the correct statement:

(a.) Each \(E_{i}\) is countable
(b.) \(\mathbb{R}-\cup E_{i}\) is uncountable for \(i=1\) to \(n\)
(c.) \(\mathbb{R}-\cup E_{i}\) is uncountable for all \(i \in \mathbb{N}\)
(d.) None of these

32. Choose the incorrect statement:

(a.) \(S \subset T \Leftrightarrow S^{\prime} \subset T^{\prime}\)
(b.) \(x \in S^{\prime} \Leftrightarrow x \in (S \cup\{x\})\) where \(S\) is any subset of \(\mathbb{R}\)
(c.) If \(S=\left\{\frac{1}{2 m}+\frac{1}{2 n} ; m, n \in \mathbb{N}\right\}, \quad\) then \(\left(S^{\prime}\right)^{\prime}=\{0\}\)
(d.) \((1,3)=\cup\left[1+\frac{1}{n}, 3-\frac{1}{n}\right] \forall n \in \mathbb{N}\)

33. Consider the following statements and choose the correct one:

(a.) Every infinite and bounded set must have a limit point
(b.) Any finite set cannot have a limit point
(c.) Any infinite but unbounded set can't have a limit point
(d.) None of these

34. Let \(A\) be a proper non-empty closed subset of \(\mathbb{R}\). Then \(A\) is:

(a.) The closure of the interior of \(A\)
(b.) a countable set
(c.) a compact set
(d.) not open

35. Let \(E \subset \mathbb{R}\), where \(E \neq \phi\). Let (1), (2), and (3) denote the following conditions:

(1) \(E\) is infinite

(2) \(E\) is bounded

(3) \(E\) is closed

Select the correct statement(s):

(a.) Condition 1 is necessary for \(E\) to have a limit point
(b.) Conditions 1 and 2 together are sufficient for \(E\) to have a limit point
(c.) Conditions 1 and 3 together are sufficient for \(E\) to have a limit point
(d.) Condition 3 is sufficient for every limit point of \(E\) to belong to \(E\)

36. Which of the following subsets of \(\mathbb{R}\) is closed?

(a.) \([0,1] \cup[2,3] \cup[4,5]\)
(b.) \([0,1]\)
(c.) \((1, \infty)\)
(d.) The set of rational numbers in \([0,1]\)

37. Which set(s) is/are not open?

(a.) \(\bigcap_{n=1}^{x} G_{n}\), where \(G_{n}=]-\frac{1}{n}, \frac{1}{n}[\)
(b.) The set \(\mathbb{Q}\) of rational numbers
(c.) The union of an arbitrary family of open sets
(d.) All of the above