{\text{Let G be a group and H be a subgroup of G then H is normal iff }}\left( {{\text{aH}}} \right)\left( {{\text{bH}}} \right) \ne \left( {{\text{ab}}} \right){\text{H}},{\text{ for all a}},{\text{b}} \in {\text{G}}.

• A) True
• B) False
• C)
• D)

{\text{The}}\,{\text{factor}}\,{\text{group}}\,Z/nZ\,{\text{is}}\,{\text{isomorphic}}\,{\text{to}}\,{Z_n}

• A) {\text{True}}
• B) {\text{False}}
• C)
• D)

{\text{A homomorphism }}\varphi :G \to G'\,is{\text{ injective iff }}\ker {\text{(}}\varphi ) = \left\{ e \right\}{\text{.}}

• A) True
• B) False
• C)
• D)

{\text{If}}\,G\,{\text{is}}\,{\text{a}}\,{\text{finite}}\,{\text{group}}\,{\text{and}}\,N\,{\text{is}}\,{\text{a}}\,{\text{proper}}\,{\text{normal}}\,{\text{subgroup}}\,{\text{of}}\,G\,{\text{then}}\,G/N\,{\text{has}}\,{\text{the}}\,{\text{same}}\,{\text{structure}}\,{\text{as}}\,G.

• A) {\text{True}}
• B) {\text{false}}
• C)
• D)

\[{\text{Let G be a group and H be a subgroup of G then H is normal iff }}\left( {{\text{aH}}} \right)\left( {{\text{bH}}} \right) \ne \left( {{\text{ab}}} \right){\text{H}},{\text{ for all a}},{\text{b}} \in {\text{G}}.\]

• A) False
• B) True
• C)
• D)

{\text{The}}\,{\text{trivial}}\,{\text{subgroup}}\,N = \{ 0\} \,{\text{of}}\,Z\,{\text{is}}\,{\text{a}}\,{\text{normal}}\,{\text{subgroup}}\,{\text{of}}\,Z.

• A) {\text{True}}
• B) {\text{False}}
• C)
• D)

Let $\left( {X,\tau } \right)$ be a metrizable then which of the following statement is true?

• A) $\left( {X,\tau } \right)$ has the countable chain collection
• B) $\left( {X,\tau } \right)$ is second countable.
• C) All of them
• D) $\left( {X,\tau } \right)$ is separable.

Which of the following statements about open balls in a metric space is true?

• A) An open ball always includes its boundary points.
• B) The centre of an open ball is not necessarily contained in the ball.
• C) Every open ball is an open set in the metric space.
• D) The radius of an open ball can be negative.

A homoemorphic map between two topological spaces is always …………..

• A) continuous
• B) an identity map
• C) piecewise continuous
• D) discontinuous

Two spaces are called topologically equivalent if there exists a ………… between two topological spaces.

• A) an injective map
• B) surjective map
• C) homeomorphism
• D) bijection

A topological property is a property of a topological space that is preserved under homeomorphisms, which of the following represent the topological property ..........

• A) All of these options
• B) Connectedness and Compactness
• C) Discreteness
• D) Countability

$${T_{X\, \times \,Y}}$$ denotes ...........on $X\, \times \,Y$.

• A) co finite topology
• B) product topology
• C) lower limit topology
• D) upper limit topology

Which of the following best describes a projection map in topology?

• A) A continuous map from a topological space to a discrete set.
• B) A map that assigns every point in a Cartesian product space to its first component.
• C) A homeomorphism between two topological spaces.
• D) A map that preserves the open sets of a topological space.

A topological space is said to be the second countable space if it has a ..............base.

• A) uncountable closed
• B) countable open
• C) countable closed
• D) uncountable open

A map ‘f’ from a topological space X to a topological space Y is ………..if and only if the image of each closed subset of X is closed in Y.

• A) a closed map
• B) a surjective map
• C) an open map
• D) an injective map

A map ‘f’ between two topological spaces is called …………… iff ‘f’ is bijective, continuous and f-1(f inverse) is continuous.

• A) automorphism
• B) isomorphism
• C) bijection
• D) homeomorphism

Every metric space is first countable.

• A) True
• B) False
• C)
• D)

Let $X$ and $Y$ be topological spaces. A map $f:X \to Y$ is called a Closed Map if________

• A) None of them
• B) for every open set $U \subseteq X$, the image $f(U) \subseteq Y$ is open.
• C) for every closed set $U \subseteq X$, the image $f(U) \subseteq Y$ is closed
• D) for every open set $U \subseteq X$, the image $f(U) \subseteq Y$ is closed.

A topological space is said to be a separable space if it has a ................subset.

• A) uncountable dense
• B) uncountable dense
• C) countable dense
• D) countable open

Which of the following properties must a relation R on a set A satisfy to be an equivalence relation?

• A) antisymmetry and transitivity
• B) symmetry and transitivity
• C) reflexivity, symmetry, and transitivity
• D) reflexivity and symmetry