CSIR NET Maths previous year question papers

Single choice · Topology

1. A

   25

2. B

   120

3. C

   24

4. D

   6

Single choice · Topology

1. A

   If X × Y is Hausdorff, then the statement S is true.

2. B

   If Y is connected, then the statement S is true.

3. C

   If X × Y is regular, then the statement S is true.

4. D

   If Y is compact, then the statement S is true.

Multi-select · full marks only for the exact set · Topology

1. A

   X is metrizable.

2. B

   X is compact.

3. C

   X is Hausdorff.

4. D

   X is connected.

Multi-select · full marks only for the exact set · Topology

1. A

   $(\mathbb{R}, \tau_E)$ is a normal space.

2. B

   Let $\tau$ be a topology on $\mathbb{R}$. If the identity function of $\mathbb{R}$ is a continuous map from $(\mathbb{R},\tau_E)$ to $(\mathbb{R},\tau)$, then $(\mathbb{R},\tau)$ is a regular space.

3. C

   Let $\tau$ be a topology on $\mathbb{R}$. If the identity function of $\mathbb{R}$ is a continuous map from $(\mathbb{R},\tau)$ to $(\mathbb{R},\tau_E)$, then $(\mathbb{R},\tau)$ is a Hausdorff space.

4. D

   $\mathbb{R}$ is a regular space in the finite-complement topology.

Single choice · Topology

1. A

   {m + ne$^(2\pi{}i/3)$ | $m, n \in \mathbb{Z}$} is a dense subset of $\mathbb{C}$.

2. B

   Open connected subsets of $\mathbb{R}^{3}$ need not be path-connected.

3. C

   Let X be a topological space and $p : X \to \mathbb{R} a$ continuous surjective open map. If p⁻$^{1}(${$\alpha$}) is connected for every $\alpha \in \mathbb{R}$, then X must be connected.

4. D

   Compact subsets of any infinite topological space are closed.

Multi-select · full marks only for the exact set · Topology

1. A

   If X and Y are compact, Y is Hausdorff and f is onto, then X is also Hausdorff.

2. B

   If X is an infinite compact set and f is a homeomorphism from X to f(X) (where f(X) is given the subspace topology), then Y is compact.

3. C

   If X is Hausdorff and f is onto, then Y is Hausdorff.

4. D

   If f is a homeomorphism, then X is second-countable if and only if Y is second-countable.

Multi-select · full marks only for the exact set · Topology

1. A

   The set of all irrational numbers is dense in $(\mathbb{R}, \tau)$.

2. B

   For each prime number p, the set {$0, \sqrt{p}$} is dense in $(\mathbb{R}, \tau)$.

3. C

   [0, 1] is compact in $(\mathbb{R}, \tau)$.

4. D

   $(\mathbb{R}, \tau)$ is Hausdorff.

Multi-select · full marks only for the exact set · Topology

1. A

   The spaces B, the closure of W, and C are homeomorphic.

2. B

   The spaces B, W, C are homeomorphic.

3. C

   If C is compact, then A, B, C are homeomorphic.

4. D

   If A is connected, then B and C are connected.

Multi-select · full marks only for the exact set · Topology

1. A

   X × [0, 1] is closed in X × Y with respect to the product topology

2. B

   X × [0, 1] is compact with respect to the product topology

3. C

   X is compact

4. D

   X × Y is compact with respect to the product topology

Multi-select · full marks only for the exact set · Topology

1. A

   $\beta$ is a basis for topology $\tau$

2. B

   $\mathbb{R}$ is compact in the topology $\tau$

3. C

   Topology $\tau$ is the same as the Euclidean topology

4. D

   Topology $\tau$ is Hausdorff

Multi-select · full marks only for the exact set · Topology

1. A

   $\mathbb{Q}^{2}$ is connected.

2. B

   If A is a non-empty connected subset of $\mathbb{Q}^{2}$, then A has exactly one element.

3. C

   $\mathbb{Q}^{2}$ is Hausdorff.

4. D

   {$(x, y) \in \mathbb{Q}^{2}$ | $x^{2} + y^{2} = 1$} is compact in the subspace topology.