Connectedness and Compactness - Topology - Mathematics Previous Year Questions

1

2009 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2009

The subspace \(\mathbb{Q} \times [0, 1]\) of \(\mathbb{R}^2\) (with the usual topology) is

A

dense in \(\mathbb{R}^2\)

B

connected

C

separable

D

compact

Open complete paper

2

2009 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2009

Consider the topology \( \tau = \{ G \subseteq \mathbb{R} : \mathbb{R} \setminus G \text{ is compact in } (\mathbb{R}, \tau_u) \} \cup \{ \phi, \mathbb{R} \} \) on \( \mathbb{R} \), where \( \tau_u \) is the usual topology on \( \mathbb{R} \) and \( \phi \) is the empty set. Then \( (\mathbb{R}, \tau) \) is

A

a connected Hausdorff space

B

connected but NOT Hausdorff

C

Hausdorff but NOT connected

D

neither connected nor Hausdorff

Open complete paper

3

2010 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2010

Let \( X = \mathbb{N} \) be equipped with the topology generated by the basis consisting of sets \( A_n = \{ n, n+1, n+2, ... \} \), \( n \in \mathbb{N} \). Then \( X \) is

A

Compact and connected

B

Hausdorff and connected

C

Hausdorff and compact

D

Neither compact nor connected

Open complete paper

4

2011 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2011

The subspace \( P = \{(x, y, z) \in \mathbb{R}^3 : z = x^2 + y^2 + 1\} \) is

A

compact and connected

B

compact but not connected

C

not compact but connected

D

neither compact nor connected

Open complete paper

5

2011 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2011

For which subspace \( X \subseteq \mathbb{R} \) with the usual topology and with \( \{0, 1\} \subseteq X \), will a continuous function \( f : X \to \{0, 1\} \) satisfying \( f(0) = 0 \) and \( f(1) = 1 \) exist?

A

\(X = [0, 1] \)

B

\(X = [-1, 1] \)

C

\(X = \mathbb{R} \)

D

\([0, 1] \not\subseteq X \)

Open complete paper

6

2012 · Mathematics · Topology · Connectedness and Compactness

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

7

2012 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2012

Let \(X = \{a,b,c\}\) and let \(\mathcal{T} = \{ \phi, \{a\}, \{b\}, \{a,b\}, X \}\) be a topology defined on \(X\). Then which of the following statements are TRUE?
P: \((X, \mathcal{T})\) is a Hausdorff space.
Q: \((X, \mathcal{T})\) is a regular space.
R: \((X, \mathcal{T})\) is a normal space.
S: \((X, \mathcal{T})\) is a connected space.

A

P and Q

B

Q and R

C

R and S

D

P and S

Open complete paper

8

2013 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2013

Let \(X\) be a compact Hausdorff topological space and let \(Y\) be a topological space. Let \(f: X \to Y\) be a bijective continuous mapping. Which of the following is TRUE?

A

\(f\) is a closed map but not necessarily an open map

B

\(f\) is an open map but not necessarily a closed map

C

\(f\) is both an open map and a closed map

D

\(f\) need not be an open map or a closed map

Open complete paper

9

2013 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2013

Which of the following subsets of \(\mathbb{R}^2\) is NOT compact?

A

\(\{(x, y) \in \mathbb{R}^2 : -1 \le x \le 1, y = \sin x\}\)

B

\(\{(x, y) \in \mathbb{R}^2 : -1 \le y \le 1, y = x^8 - x^2 - 1\}\)

C

\(\{(x, y) \in \mathbb{R}^2 : y = 0, \sin(e^{-x}) = 0\}\)

D

\(\{(x, y) \in \mathbb{R}^2 : x > 0, y = \sin(\frac{1}{x})\} \cap \{(x, y) \in \mathbb{R}^2 : x > 0, y = \frac{1}{x}\}\)

Open complete paper

10

2013 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2013

Let \(S = \{z \in \mathbb{C} : |z| = 1\}\) with the induced topology from \(\mathbb{C}\) and let \(f: [0, 2] \to S\) be defined as \(f(t) = e^{2\pi i t}\). Then, which of the following is TRUE?

A

\(K\) is closed in \([0, 2] \Rightarrow f(K)\) is closed in \(S\)

B

\(U\) is open in \([0, 2] \Rightarrow f(U)\) is open in \(S\)

C

\(f(X)\) is closed in \(S \Rightarrow X\) is closed in \([0, 2]\)

D

\(f(Y)\) is open in \(S \Rightarrow Y\) is open in \([0, 2]\)

Open complete paper

11

2013 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2013

Let \(q_1, ..., q_{n_0+1}\) be \(n_0 + 1\) distinct points and \(Y = X \setminus \{q_1, ..., q_{n_0+1}\}\). Let \(m\) be the number of connected components of \(Y\). The maximum possible value of \(m\) is ______

Check answer

Open complete paper

12

2014 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2014

Let \(X = [0,1) \cup (1,2)\) be the subspace of \(\mathbb{R}\), where \(\mathbb{R}\) is equipped with the usual topology. Which of the following is FALSE?

A

\(A\) There exists a non-constant continuous function \(f: X \to \mathbb{Q}\)

B

\(B\) \(X\) is homeomorphic to \((-\infty,-3) \cup [0,\infty)\)

C

\(C\) There exists an onto continuous function \(f: [0,1] \to \bar{X}\), where \(\bar{X}\) is the closure of \(X\) in \(\mathbb{R}\)

D

\(D\) There exists an onto continuous function \(f: [0,1] \to X\)

Open complete paper

13

2016 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2016

Let (ℝ, τ) be a topological space with the cofinite topology. Every infinite subset of ℝ is

A

Compact but NOT connected

B

Both compact and connected

C

NOT compact but connected

D

Neither compact nor connected

Open complete paper

14

2016 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2016

Let \(\mathbb{M}\) be the set of all \(n \times n\) real matrices with the usual norm topology. Consider the following statements P and Q:
(P) : The set of all symmetric positive definite matrices in \(\mathbb{M}\) is connected.
(Q) : The set of all invertible matrices in \(\mathbb{M}\) is compact.
Which of the above statements hold TRUE?

A

both P and Q

B

only P

C

only Q

D

Neither P nor Q

Open complete paper

15

2017 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2017

Consider the topology \(\mathcal{T} = \{ U \subseteq \mathbb{Z} : \mathbb{Z} \setminus U \text{ is finite or } 0 \notin U \}\) on \(\mathbb{Z}\). Then, the topological space \((\mathbb{Z}, \mathcal{T})\) is

A

compact but NOT connected

B

connected but NOT compact

C

both compact and connected

D

neither compact nor connected

Open complete paper

16

2018 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2018

Let \(X\) and \(Y\) be metric spaces and let \(f : X \to Y\) be a continuous map. For any subset \(S\) of \(X\), which one of the following statements is true?

A

If \(S\) is open, then \(f(S)\) is open

B

If \(S\) is connected, then \(f(S)\) is connected

C

If \(S\) is closed, then \(f(S)\) is closed

D

If \(S\) is bounded, then \(f(S)\) is bounded

Open complete paper

17

2018 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2018

Let \(f : X \to Y\) be a continuous map from a Hausdorff topological space \(X\) to a metric space \(Y\). Consider the following two statements:
P: \(f\) is a closed map and the inverse image \(f^{-1}(y) = \{x \in X : f(x) = y\}\) is compact for each \(y \in Y\).
Q: For every compact subset \(K \subset Y\), the inverse image \(f^{-1}(K)\) is a compact subset of \(X\).
Which one of the following is true?

A

Q implies P but P does NOT imply Q

B

P implies Q but Q does NOT imply P

C

P and Q are equivalent

D

neither P implies Q nor Q implies P

Open complete paper

18

2020 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2020

Let \( P(\mathbb{R}) \) denote the power set of \( \mathbb{R} \), equipped with the metric
\( d(U, V) = \sup_{x \in \mathbb{R}} |\chi_U(x) - \chi_V(x)| \),
where \( \chi_U \) and \( \chi_V \) denote the characteristic functions of the subsets \( U \) and \( V \), respectively, of \( \mathbb{R} \). The set \( \{ m : m \in \mathbb{Z} \} \) in the metric space \( (P(\mathbb{R}), d) \) is

A

bounded but not totally bounded

B

totally bounded but not compact

C

compact

D

not bounded

Open complete paper

19

2020 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2020

Suppose that
\[ X = \{(0,0)\} \cup \left\{ \left(x, \sin \frac{1}{x}\right) : x \in \mathbb{R} \setminus \{0\} \right\} \]
and
\[ Y = \{(0,0)\} \cup \left\{ \left(x, x \sin \frac{1}{x}\right) : x \in \mathbb{R} \setminus \{0\} \right\} \]
are metric spaces with metrics induced by the Euclidean metric of \(\mathbb{R}^2\). Let \(B_X\) and \(B_Y\) be the open unit balls around \((0,0)\) in \(X\) and \(Y\), respectively. Consider the following statements:
I : The closure of \(B_X\) in \(X\) is compact.
II : The closure of \(B_Y\) in \(Y\) is compact.
Then

A

both I and II are true

B

I is true but II is false

C

I is false but II is true

D

both I and II are false

Open complete paper

20

2020 · Mathematics · Topology · Connectedness and Compactness

Mathematics (MA) 2020

Suppose that \( U = \mathbb{R}^2 \setminus \{(x,y) \in \mathbb{R}^2 : x, y \in \mathbb{Q}\} \), \( V = \mathbb{R}^2 \setminus \{(x,y) \in \mathbb{R}^2 : x > 0, y = \frac{1}{x}\} \). Then, with respect to the Euclidean metric on \( \mathbb{R}^2 \),

A

both \(U\) and \(V\) are disconnected

B

\(U\) is disconnected but \(V\) is connected

C

\(U\) is connected but \(V\) is disconnected

D

both \(U\) and \(V\) are connected

Open complete paper