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

Practice IIT JAM Mathematics Test Series for Free only On www.examflame.com and take your preparation to the another level. In the Series of Tests for IIT JAM 2023 this is the test of 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. Ei ∩ Ei‘= Then select the correct statement

(a) Each Ei is countable
(b) R-∪Ei is uncountable for i =1 to n
(c) R-∪Ei 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 Qc 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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