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**topic Point Set Topology**of the

**Chapter Real Analysis.**These Questions are prepared as per the Latest Syllabus of IIT JAM Mathematics 2023 . Practicing mock tests/Test series, you will get an idea about how and which type of Question will ask in the Examination. It also boost your confidence level. Also Solve

**IIT JAM Mathematics Previous Year Question Paper**. And Don’t Forget to Shore with Your Friends.

**IIT JAM Mathematics Test Series 2023 : Real Analysis – Point Set Topology**

**01. Which of the following subsets of R is closed?**

(a) [0,1] U [2, 3] U [4, 5]

(b) [0, 1]

(c) (1, ∞)

(d.) The set of rational numbers in [0,1]

**02. Let G and H be nonempty subsets of R where G is connected and G∪H is not connected. Which one of the following statements is true for all such G and H?**

(a) If G∩H = Ø, then His connected

(b) If G∩H = Ø, then H is not connccted

(c) If G∩H = Ø, then H is connected

(d) If G∩H = Ø, then H is not connected

**03. Let S be a nonempty subset of R. If it is a finite union of disjoint bounded intervals, then which one of the following is true?**

(a) If S is not compact, then sup S ∉ S and infS ∉ S .

(b) Even if sup S ∈ S and infS ∈ S, S need not be compact.

(c) If sup S ∈ S and infS ∈ S,then S is compact

(d) Even if Sis compact, it is not necessary that sup S ∈ S and infS ∈ S

**04. Let E are subsets of R such that. E _{i} ∩ E_{i}‘= Then select the correct statement**

(a) Each E_{i} is countable

(b) R-∪E_{i} is uncountable for i =1 to n

(c) R-∪E_{i} is uncountable ∀**i **∈ N

(d) None of these.

**05. Choose the incorrect statement**:

(a) S ⊂ T ⇔ S’ ⊂ T’

(b) x ∈ S’ ⇔ x ∈ (S∪{r})’ where S is any subset of R

(c) If S = {^{1}/_{2m} + ^{1}/_{2n} ; m, n ∈ N} then (S’)’ = {0}

(d) (1,3) = ∪[1+^{1}/_{n} , 3-^{1}/_{n}] ∀n ∈ N

**06. Consider the following statements and choose corect**

(a) Every infinite and bounded set must have a limit point

(b) Any finite set cannot have a limit point

(c) Any infinite but unboundediset can’t have a limit point

(d) None of these

**07. Let A be a closed subset of R, A ≠ Ø, R. Then A is**

(a) the closure ofthe interior of A

(b) a countable set.

(c) a compact set

(d) not open.

**08. Let E ⊂ R, E ≠ Φ. Let (1), (2) and (3) denote the following conditions:****E is infinite****E is bounded****E is closed**

(a) 1 is necessary for E to have a limit point

(b) 1 and 2 together are sufficient for E to have a limit point

(c) 1 and 3 together are sufficient for E to have a limit point

(d) 3 is sufñicient for every limit point of E to belong to E

**09. Let A = {m + n√2: m, n ∈ Z}, where Z stands for the set of all integers**

(a) A is dense in R

(b) A has only countable many limit point in

(c) A has no limit point in R

(d) Only irrational numbers can be limit points of A

**10. Let X= { ^{1}/_{n} : n ∈ Z, n ≥ 1} and let X’ be its closure. Then**

(a) X’\X is a single point

(b) X’\X is open in R

(c) X’\X is infinite but not open in R

(d) X’\X = Φ

**11. Which of the following sets satisfy the condition that for every positive integer n there is some a in A such that a < n?**

(a) A = {-1,5}

(b) A is empty set

(c) A = N

(d) A = {x ∈ R | x > 10}

**12. Let s be an infinite subset of R such that S∩Q = Φ Which of the following stafement is true?**

(a) S must have a limit point which belongs to Q

(b) S must have a limit pomt which belongs to R\Q

(c) S cannot be a closed set in R

(d) R/S must have a limit point which belongs to S

**13. Let S be an uncountable set and T be a set of those real number x s.t. (x-δ,x+δ)∩S set is uncountable then which of the statements is/are correct**

(a) T is countable

(b) S-T is countable

(c) S∩T is uncountable

(d) All are correct

**14. Which of the following is not a ****nbd**** of each of its point?**

(a) Set Q of rational numbers

(b) Set Q^{c} of irational mumber

(c) Set Z of integers

(d) None of these

**15. S be subset of R and infS = sup S, Then**

(a) S empty

(b) S singleton

(c) S finite but may not be singleton

(d) Can’t say

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*Answer Key*

*Answer Key*

01. | (a), (b) | 02. | (d) | 03. | (b) | 04. | (a), (b), (c) | 05. | (a) |

06. | (a), (b) | 07. | (d) | 08. | (a), (b), (d) | 09. | (a) | 10. | (a) |

11. | (a) | 12. | (d) | 13. | (b), (c) | 14. | (a), (b), (c) | 15. | (b) |

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