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

$${\text{Metric topology induced by }}d(x,y) = |x - y{\text{| on }}\mathbb{R}{\text{ is called\_\_\_\_\_}}$$

• A) indiscrete topology
• B) usual topology
• C) discrete topology
• D) None of them

A first countable space is a topological space in which there exist a..................... base at each of its point.

• A) countable local
• B) uncountable non-local
• C) countable non-local
• D) uncountable local

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

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

Let X and Y be two topological spaces. If f is a function from X to Y, then f is said to be homeomorphism if..............

• A) all of these options
• B) f is continuous
• C) f is open
• D) f is bijective

A topological space (X,ͳ) is said to be metrizable if there is a metric d on X that generate.... ...

• A) none of these
• B) both X and ͳ
• C) whole space X
• D) topology ͳ

Let$X = R$(Set of real numbers) be a usual metric space and $N \subseteq R$, then which of the following is NOT true about $N$ ?

• A) It is not a neighbourhood of any of its point.
• B) None of them.
• C) Neither of its point is an interior point.
• D) It must be an open set.