Topology (MTH634)

1. Which of the following statement is not true?

   1. Indiscrete topology is weaker than any other topology defined on the same non empty set.
   2. Discrete topology is finer than any other topology defined on the same non empty set.
   3. Discrete topology is finer than the indiscrete topology defined on the same non empty set.
   4. Indiscrete topology is finer than any other topology defined on the same non empty set.

2. The set of ________ of R (Real line) forms a topology called usual topology.

   1. all open intervals
   2. all open discs
   3. all open balls
   4. all open spheres

3. Let X = {1, 2, 3}, then P(X) = ________

   1. {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}
   2. {φ, {1}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
   3. {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, X}
   4. {φ, X}

4. If X is a finite set then co-finite topology on X is ________.

   1. indiscrete topology
   2. discrete topology
   3. lower limit topology
   4. None of the given

5. \(\bigcap\limits_{n\epsilon N} (- {1 \over n}, {1 \over n} )= \) ________, where N stands for set of natural numbers

   1. {0}
   2. {-1, 1}
   3. {}
   4. (-infinity, +infinity)

6. Which of the following are NOT topologically equal?

   1. A circle and a square
   2. A triangle and a rectangle
   3. A rectangle and a square
   4. A circle and a line

7. Let \(X = \Bbb{R}\) with usual topology and consider \(B = \{1, {1\over2}, {1\over 3}, {1\over4}, . . .\}\). The limit point of B is:

   1. 1
   2. 1⁄2
   3. 0
   4. None of the given

8. If \(X = R\) with usual Topology and consider \(B = \{1/n \text{ where n belongs to the set of Natural numbers}\}\), then the limit point of B is ________.

   1. 1
   2. 0
   3. 1⁄2
   4. 1⁄3

9. The largest topology defined on some set is the ________ topology.

   1. discrete
   2. indiscrete
   3. lower limit
   4. co finite

10. The smallest topology one can define on some set is called:

    1. indiscrete topology
    2. discrete topology
    3. comparable topology
    4. usual topology

11. Let \(X = \{a, b, c, d\}\) and \(τ = \{φ, \{c\}, \{a, c\}, \{b, c, d\}, X\}\) be a topology on X. The closed set in X is:

    1. {b, d}
    2. {c}
    3. {d}
    4. None of the given

12. Let X = {a, b, c}. The following set is a topology on X.

    1. {φ, {b}, {c}, X}
    2. {φ, {a}, {b}, X}
    3. {φ, {a}, {b, c}, X}
    4. None of the given

13. Topology can be a useful tool in those problems where ________ study is more effective.

    1. Qualitative
    2. Quantitative

14. Let \(X = \Bbb{R}\) with usual topology and \(A = (0, 3)\). The limit point of A is:

    1. 1.5
    2. 4
    3. 5
    4. 0.5

15. If X is finite and has n elements then power set of X has ________ elements.

    1. \(2^n\)
    2. \(2^{n-1}\)
    3. \(2^{n+1}\)
    4. None of the given

16. Let X = {a, b, c, d}. The following set represents a topology on X.

    1. {φ, {a}, {a, b}, X}
    2. {φ, {a}, {b}, X}
    3. {φ, {a}, {b, c}, X}
    4. {φ, {c}, {a, b}, X}

17. Let τ be a topology on X. The elements of τ are called:

    1. closed set
    2. open sets
    3. derived set
    4. dense set

18. Only one topology can be defined on a set.

    1. True
    2. False

19. If one shape can be deformed in another shape then topologically they are considered to be ________.

    1. the same
    2. dissimilar

20. The set of all open intervals of R is a topology on R, called

    1. discrete topology
    2. cofinite topology
    3. real topology
    4. usual topology

21. Topology means the study of something with respect to its ________.

    1. place
    2. volume
    3. perimeter
    4. area

22. Let X = {a, b, c, d}. The following set is not a topology on X.

    1. {φ, {a}, {a, b}, X}
    2. {φ, {a}, {b}, X}
    3. {φ, {a}, {a, c}, X}
    4. None of the given

23. Let X = {a, b, c, d}. The following set is a topology on X.

    1. {φ, {a}, {b}, {c}, X}
    2. {φ, {c, d}, {b, c, d}, X}
    3. {φ, {a}, {b}, X}
    4. None of the given

24. Let \(X = \{a, b, c, d\}\). The following set is a topology on X.

    1. {φ, {b}, {c}, X}
    2. {φ, {b}, {a, b}, {c}, X}
    3. {φ, {b, d}, {a, b, c}, {b}, X}
    4. None of the given

25. Which of the following statement is true?

    1. In both Geometry and Topology we care about exact measurements.
    2. In Geometry we don't care about measurements but in Topology we care about exact measurements.
    3. In Geometry we care about measurements but in Topology we don't care about exact measurements.
    4. In both Geometry and Topology we don't care about exact measurements.

26. Which of the following topology is called "Finite Complement Topology"?

    1. Discrete Topology
    2. Indiscrete Topology
    3. Co finite Topology
    4. Lower Limit Topology

27. In a topological space the intersection of any collection of closed sets is ________.

    1. Closed
    2. Open
    3. Neither open nor closed
    4. None of the given

28. If in a topology τ on X, all subsets of X are called open and closed, then τ is called:

    1. discrete space
    2. indiscrete space
    3. metric space
    4. None of the given

29. Which of the following topology contains the complete power set of a set?

    1. Indiscrete topology
    2. lower limit topology
    3. Discrete topology
    4. co finite topology

30. The collection τ of subsets of X consisting of the empty set φ and all subsets of X whose complements are finite is called:

    1. discrete topology
    2. cofinite topology
    3. indiscrete topology
    4. None of the given

31. For an open ball centered at x = (x1, x2, . . ., xn) belongs to Rn, which of the following is the correct representation?

    1. \(\{y| \sqrt{\sum_{i=1}^n (x_i - y_i)^2} \lt r \} \subset R^n, \text{ where } y = (y_1, y_2, . . . , y_n) \in R^n \)
    2. \(\{y| \sqrt{\sum_{i=1}^n (x_i - y_i)^2} \le r \} \subset R^n, \text{ where } y = (y_1, y_2, . . . , y_n) \in R^n \)
    3. \(\{y| \sqrt{\sum_{i=1}^n (x_i - y_i)^2} \gt r \} \subset R^n, \text{ where } y = (y_1, y_2, . . . , y_n) \in R^n \)
    4. \(\{y| \sqrt{\sum_{i=1}^n (x_i - y_i)^2} \ge r \} \subset R^n, \text{ where } y = (y_1, y_2, . . . , y_n) \in R^n \)

Select a course code for Objective Questions:

ACC311 ACC501 BIF101 BIF401 BIF501 BIO101 BIO201 BIO202 BIO203 BIO204 BIO301 BIO302 BIO303 BIO401 BIO503 BNK603 BT101 BT201 BT301 BT302 BT401 BT402 BT403 BT404 BT405 BT406 BT501 BT504 BT505 BT604 CHE201 CHE301 CS001 CS101 CS201 CS202 CS204 CS301 CS302 CS304 CS311 CS401 CS402 CS403 CS408 CS501 CS502 CS504 CS506 CS508 CS601 CS602 CS604 CS605 CS607 CS609 CS610 CS614 CS707 CS710 ECO401 ECO402 ECO403 ECO404 EDU302 EDU401 EDU402 EDU501 EDU601 ENG101 ENG201 ENG301 ENG501 FIN611 FIN624 FIN625 ISL201 IT430 MCM101 MCM301 MGMT623 MGT101 MGT211 MGT301 MGT401 MGT402 MGT501 MGT502 MGT503 MGT510 MGT601 MGT603 MKT501 MKT610 MKT630 MTH101 MTH202 MTH302 MTH501 MTH601 MTH634 PAK301 PHY101 PHY301 PSY101 PSY403 PSY502 PSY512 SOC101 SOC301 SOC401 STA301 STA630 STA641 ZOO102 ZOO401 ZOO501 ZOO502 ZOO503 ZOO504 ZOO505 ZOO506 ZOO510 ZOO630

Select a course code for Subjective Questions:

ACC311 CS001 CS101 CS201 CS301 CS607 CS701 CS702 CS703 CS704 CS707 CS708 CS710 CS711 CS718 CS721 CS724 ECO401 ECO403 ENG101 ENG301 MGT703 MTH501 STA301