Topological Spaces and Constructions - Topology - Mathematics Previous Year Questions

1

2008 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2008

Let \(X\) be a non-empty set. Let \(\mathcal{T}_1\) and \(\mathcal{T}_2\) be two topologies on \(X\) such that \(\mathcal{T}_1\) is strictly contained in \(\mathcal{T}_2\). If \(I : (X, \mathcal{T}_2) \to (X, \mathcal{T}_1)\) is the identity map, then

A

both \(I\) and \(I^{-1}\) are continuous

B

both \(I\) and \(I^{-1}\) are not continuous

C

\(I\) is continuous but \(I^{-1}\) is not continuous

D

\(I\) is not continuous but \(I^{-1}\) is continuous

Open complete paper

2

2008 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2008

Let \(X = \mathbb{R}\) and let \(\mathfrak{I} = \{U \subseteq X : X - U \text{ is finite}\} \cup \{\phi, X\}\). The sequence \(1, \frac{1}{2}, \frac{1}{3}, \cdots, \frac{1}{n}, \cdots\) in \((X, \mathfrak{I})\)

A

converges to 0 and not to any other point of \(X\)

B

does not converge to 0

C

converges to each point of \(X\)

D

is not convergent in \(X\)

Open complete paper

3

2008 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2008

Let \(X = \{1, 2, 3\}\) and \(\mathfrak{I} = \{\phi, \{1\}, \{2\}, \{1, 2\}, \{2, 3\}, \{1, 2, 3\}\}\). The topological space \((X, \mathfrak{I})\) is said to have the property P if for any two proper disjoint closed subsets \(Y\) and \(Z\) of \(X\), there exist disjoint open sets \(U\), \(V\) such that \(Y \subseteq U\) and \(Z \subseteq V\). Then the topological space \((X, \mathfrak{I})\)

A

is \(T_1\) and satisfies P

B

is \(T_1\) and does not satisfy P

C

is not \(T_1\) and satisfies P

D

is not \(T_1\) and does not satisfy P

Open complete paper

4

2008 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2008

Then

A

\(F\) and \(f\) are closed maps

B

\(F\) is a closed map and \(f\) is not a closed map

C

\(F\) is not a closed map and \(f\) is a closed map

D

neither \(F\) nor \(f\) is a closed map

Open complete paper

5

2009 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2009

Let \( \tau_1 = \{ G \subseteq \mathbb{R} : G \text{ is finite or } \mathbb{R} \setminus G \text{ is finite} \} \) and \( \tau_2 = \{ G \subseteq \mathbb{R} : G \text{ is countable or } \mathbb{R} \setminus G \text{ is countable} \} \). Then

A

neither \(\tau_1 \) nor \(\tau_2 \) is a topology on \(\mathbb{R} \)

B

\(\tau_1 \) is a topology on \(\mathbb{R} \) but \(\tau_2 \) is NOT a topology on \(\mathbb{R} \)

C

\(\tau_2 \) is a topology on \(\mathbb{R} \) but \(\tau_1 \) is NOT a topology on \(\mathbb{R} \)

D

both \(\tau_1 \) and \(\tau_2 \) are topologies on \(\mathbb{R} \)

Open complete paper

6

2010 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2010

Let \( X = \mathbb{R} \) equipped with the topology generated by open intervals of the form \( (a, b) \) and sets of the form \( (a, b) \cap \mathbb{Q} \). Then which one of the following statements is correct?

A

\(X \) is regular

B

\(X \) is normal

C

\(X \setminus \mathbb{Q} \) is dense in \(X \)

D

\(\mathbb{Q} \) is dense in \(X \)

Open complete paper

7

2010 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2010

In the space \( X \),

A

\(P \) is closed but not open

B

\(P \) is open but not closed

C

\(P \) is both open and closed

D

\(P \) is neither open nor closed

Check answer

Open complete paper

8

2010 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2010

The boundary of \( P \) in \( X \) is

A

an empty set

B

a singleton set

C

\(P \)

D

\(X \)

Open complete paper

9

2011 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2011

Let \( P = (0,1), Q = [0,1), U = (0,1], S = [0,1], T = \mathbb{R} \) and \( A = \{P,Q,U,S,T\} \). The equivalence relation 'homeomorphism' induces which one of the following as the partition of A?

A

\{P,Q,U,S\}, \{T\}

B

\{P,T\}, \{Q\}, \{U\}, \{S\}

C

\{P,T\}, \{Q,U,S\}

D

\{P,T\}, \{Q,U\}, \{S\}

Open complete paper

10

2011 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2011

Suppose \( X \) is a finite set with more than five elements. Which of the following is TRUE?

A

There is a topology on \(X \) which is \(T_3 \)

B

There is a topology on \(X \) which is \(T_3 \) but not \(T_2 \)

C

There is a topology on \(X \) which is \(T_1 \) but not \(T_2 \)

D

There is no topology on \(X \) which is \(T_1 \)

Check answer

Open complete paper

11

2012 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2012

In a topological space, which of the following statements is NOT always true :

A

Union of any finite family of compact sets is compact.

B

Union of any family of closed sets is closed.

C

Union of any family of connected sets having a non empty intersection is connected.

D

Union of any family of dense subsets is dense.

Open complete paper

12

2012 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2012

Consider the following statements:
P: The family of subsets \(\left\{ A_n = \left( -\frac{1}{n}, \frac{1}{n} \right], n = 1,2, \dots \right\}\) satisfies the finite intersection property.
Q: On an infinite set \(X\), a metric \(d : X \times X \to R\) is defined as \(d(x,y) = \begin{cases} 0, & x = y \\ 1, & x \neq y \end{cases}\). The metric space \((X,d)\) is compact.
R: In a Frechet (\(T_1\)) topological space, every finite set is closed.
S: If \(f : R \to X\) is continuous, where \(R\) is given the usual topology and \((X, \tau)\) is a Hausdorff (\(T_2\)) space, then \(f\) is a one-one function.
Which of the above statements are correct?

A

P and R

B

P and S

C

R and S

D

Q and S

Open complete paper

13

2013 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2013

Consider \(\mathbb{R}^2\) with the usual topology. Which of the following statements are TRUE for all \(A, B \subseteq \mathbb{R}^2\)?
P:\(A \cup B = \overline{A} \cup \overline{B}\).
Q:\(A \cap B = \overline{A} \cap \overline{B}\).
R:\((A \cup B)^{\circ} = A^{\circ} \cup B^{\circ}\).
S:\((A \cap B)^{\circ} = A^{\circ} \cap B^{\circ}\).

A

P and R only

B

P and S only

C

Q and R only

D

Q and S only

Open complete paper

14

2014 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2014

Let \(X\) be a set with at least two elements. Let \(\tau\) and \(\tau'\) be two topologies on \(X\) such that \(\tau' \neq \{\phi, X\}\). Which of the following conditions is necessary for the identity function \(id : (X, \tau) \to (X, \tau')\) to be continuous?

A

\(\tau \subseteq \tau'\)

B

\(\tau' \subseteq \tau\)

C

no conditions on \(\tau\) and \(\tau'\)

D

\(\tau \cap \tau' = \{\phi, X\}\)

Open complete paper

15

2014 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2014

Let d₁, d₂ and d₃ be metrics on a set X with at least two elements. Which of the following is NOT a metric on X?

A

min(d₁, 2)

B

max(d₂, 2)

C

d₃/(1+d₃)

D

(d₁+d₂+d₃)/3

Open complete paper

16

2014 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2014

Consider the partial order in \(\mathbb{R}^2\) given by the relation \((x_1, y_1) < (x_2, y_2)\) EITHER if \(x_1 < x_2\) OR if \(x_1 = x_2\) and \(y_1 < y_2\). Then in the order topology on \(\mathbb{R}^2\) defined by the above order

A

[0,1] × {1} is compact but [0,1] × [0,1] is NOT compact

B

[0,1] × [0,1] is compact but [0,1] × {1} is NOT compact

C

both [0, 1] × {0, 1} and [0,1] × {1} are compact

D

both [0, 1] × {0, 1} and [0,1] × {1} are NOT compact

Open complete paper

17

2017 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2017

Let \(X\) and \(Y\) be topological spaces and let \(f : X \to Y\) be a continuous surjective function. Which one of the following statements is TRUE?

A

If \(X\) is separable, then \(Y\) is separable

B

If \(X\) is first countable, then \(Y\) is first countable

C

If \(X\) is Hausdorff, then \(Y\) is Hausdorff

D

If \(X\) is regular, then \(Y\) is regular

Open complete paper

18

2017 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2017

Let \( \mathcal{T}_u \) and \( \mathcal{T}_d \) denote the usual topology and the discrete topology on \( \mathbb{R} \), respectively. Consider the following three topologies:
\( \mathcal{T}_1 = \) Usual topology on \( \mathbb{R}^2 = \mathbb{R} \times \mathbb{R} \),
\( \mathcal{T}_2 = \) Topology generated by the basis \( \{U \times V : U \in \mathcal{T}_d, V \in \mathcal{T}_u\} \) on \( \mathbb{R} \times \mathbb{R} \),
\( \mathcal{T}_3 = \) Dictionary order topology on \( \mathbb{R} \times \mathbb{R} \).
Then

A

\(\mathcal{T}_3 \subseteq \mathcal{T}_1 \subseteq \mathcal{T}_2 \)

B

\(\mathcal{T}_1 \subseteq \mathcal{T}_2 \subseteq \mathcal{T}_3 \)

C

\(\mathcal{T}_3 \subseteq \mathcal{T}_2 \subseteq \mathcal{T}_1 \)

D

\(\mathcal{T}_1 \subseteq \mathcal{T}_2 = \mathcal{T}_3 \)

Open complete paper

19

2018 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2018

Let \(X\) denote \(\mathbb{R}^2\) endowed with the usual topology. Let \(Y\) denote \(\mathbb{R}\) endowed with the co-finite topology. If \(Z\) is the product topological space \(Y \times Y\), then

A

the topology of \(X\) is the same as the topology of \(Z\)

B

the topology of \(X\) is strictly coarser (weaker) than that of \(Z\)

C

the topology of \(Z\) is strictly coarser (weaker) than that of \(X\)

D

the topology of \(X\) cannot be compared with that of \(Z\)

Open complete paper

20

2018 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2018

Consider \(\mathbb{R}^n\) with the usual topology for \(n = 1, 2, 3\). Each of the following options gives topological spaces \(X\) and \(Y\) with respective induced topologies. In which option is \(X\) homeomorphic to \(Y\)?

A

\(X = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1\}, \quad Y = \{(x,y,z) \in \mathbb{R}^3 : z = 0, x^2 + y^2 \neq 0\}\)

B

\(X = \{(x,y) \in \mathbb{R}^2 : y = \sin(1/x), 0 < x \leq 1\} \cup \{(x,y) \in \mathbb{R}^2 : x = 0, -1 \leq y \leq 1\}, \quad Y = [0,1] \subseteq \mathbb{R}\)

C

\(X = \{(x,y) \in \mathbb{R}^2 : y = x \sin(1/x), 0 < x \leq 1\}, \quad Y = [0,1] \subseteq \mathbb{R}\)

D

\(X = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1\}, \quad Y = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = z^2 \neq 0\}\)

Open complete paper