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

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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 \(X\)

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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

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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

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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 \(\mathbb{R} \)

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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 \(X \)

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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

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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

\(X \)

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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\}

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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

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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.

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12

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

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13

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

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

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14

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

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15

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

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16

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

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

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17

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 \(Z\)

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18

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

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

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19

2019 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2019

Let \(T_1\) be the co-countable topology on \(\mathbb{R}\) (the set of real numbers) and \(T_2\) be the co-finite topology on \(\mathbb{R}\). Consider the following statements: I. In \((\mathbb{R}, T_1)\), the sequence \(\left\{ \frac{1}{n} \right\}_{n=1}^{\infty}\) converges to 0. II. In \((\mathbb{R}, T_2)\), the sequence \(\left\{ \frac{1}{n} \right\}_{n=1}^{\infty}\) converges to 0. III. In \((\mathbb{R}, T_1)\), there is no sequence of rational numbers which converges to \(\sqrt{3}\). IV. In \((\mathbb{R}, T_2)\), there is no sequence of rational numbers which converges to \(\sqrt{3}\). Which of the above statements are TRUE?

A

I and II only

B

II and III only

C

III and IV only

D

I and IV only

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20

2019 · Mathematics · Topology · Topological Spaces and Constructions

Mathematics (MA) 2019

Consider the ordered square \( I_{o}^{2} \), the set \( [0,1] \times[0,1] \) with the dictionary order topology. Let the general element of \( I_{o}^{2} \) be denoted by \( x \times y \), where \( x, y \in[0,1] \). Then the closure of the subset \( S=\left\{x \times \frac{3}{4}: 0

A

\(S \bigcup\left(\{a, b\} \times\{0\}\right) \bigcup\left(\{a, b\} \times\{1\}\right) \)

B

\(S \bigcup\left(\{a, b\} \times\{0\}\right) \bigcup\left(\{a, b\} \times\{1\}\right) \)

C

\(S \bigcup\left(\{a, b\} \times\{0\}\right) \bigcup\left(\{a, b\} \times\{1\}\right) \)

D

\(S \bigcup\left(\{a, b\} \times\{0\}\right) \)

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