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

• 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.

A topological space X is said to be separable if it has a countable dense subset. Which of the following is a countable dense subset of the real numbers?

• A) The set of natural numbers
• B) The set of rational numbers
• C) The set of all real numbers
• D) The set of integers

Which of the following statements best defines a metric space?

• A) A set that is equipped with a topology.
• B) A set where every element has a unique inverse element.
• C) A set with a binary operation that combines any two elements to form a third element.
• D) A set with a function that measures the distance between any two elements in the set.

If $(X,d)$ is a compact metric space, then______

• A) every sequence which has a convergent subsequence is a Cauchy sequence.
• B) every sequence is a Cauchy sequence.
• C) every sequence has a subsequence which is a Cauchy sequence.
• D) every Cauchy sequence converges.

Let$X = R$(Set of real numbers) be a usual metric space and $N \subseteq R$, then which of the following is true about $N$ ?

• A) It is not an open set.
• B) All of them.
• C) None of its point is an interior point.
• D) It has disjoint intersection with the set of its interior points.

What is an open cover of a topological space X?

• A) A collection of open sets that are disjoint.
• B) A single open set that contains X.
• C) A collection of closed sets whose intersection is X.
• D) A collection of open sets whose union is X.

If $X$ has more than two points and $\left( {X,\tau } \right)$ be an indiscrete topology then which of the following statement is true about $\left( {X,\tau } \right)$ ?

• A) It is Haussdorff.
• B) It is metrizable.
• C) None of them.
• D) It is not metrizable.

Let $X = \left\{ {1,2,3,4,5,6} \right\}$ and $\tau = \left\{ {\emptyset ,\{ 1\} ,\{ 2\} ,\{ 1,2\} ,X} \right\}$ be a topology on $X$, then the local base ( ${B_x}$ ) of the point $x = 1$ is_______

• A) $\left\{ {\{ 1\} ,\{ 2\} ,X} \right\}$.
• B) $\left\{ {\{ 1\} ,\{ 2\} ,\{ 1,2\} ,X} \right\}$.
• C) $\left\{ {\{ 1\} ,\{ 1,2\} ,X} \right\}$.
• D) None of them

Let $X = \left\{ {1,2,3,4} \right\}$ and $\tau = \left\{ {\emptyset ,\{ 1\} ,\{ 2\} ,\{ 1,2\} ,X} \right\}$ be a topology on $X$, then which of the following is NOT true ?

• A) $\left( {X,\tau } \right)$ be a topological space.
• B) Every element of $X$ has uncountable local base.
• C) The local base of the element 4 is $\emptyset$.
• D) $\left( {X,\tau } \right)$ be a first countable space.

Which of the following statements correctly characterizes a T1 topological space?

• A) Every singleton subset is open.
• B) Every subset is closed.
• C) Every subset is open.
• D) Every singleton subset is closed.

Given a Hausdorff space $(X, T)$ and a subset $Y \subset X$, with the subspace topology $(Y, T_{Y})$, which of the following statements is true?

• A) Y must be compact
• B) Y with the subspace topology $(Y, T_{Y})$ is also Hausdorff.
• C) Y must be connected.
• D) Y with the subspace topology $(Y, T_{Y})$ is not Hausdorff.

Which of the following is an example of a metric space that is also Hausdorff?

• A) The discrete metric space on any set.
• B) The set of complex numbers $\mathbb{C}$ with the metric defined on the set of complex numbers.
• C) All of these
• D) The set of real numbers $\mathbb{R}$ with the standard Euclidean metric.

Which of the following statements is true about discrete topological space?

• A) None of these
• B) Every discrete topological space is regular.
• C) Every discrete topological space may or may not be regular.
• D) Every discrete topological space is not regular.

Which of the following space is a Housdorff?

• A) A non empty set X with indiscrete space.
• B) An Infinite set with cofinite toplogy.
• C) Real numbers with toplogy generated by { ${{(a, \infty) | a \in R}}$}
• D) Real numbers with Lower limit topology.

Which of the following statements is equivalent to the statement that a topological space X is T1?

• A) For any two distinct points x and y in X, there exists an open set V such that y ∈ V and x ∉ V.
• B) For any two distinct points x and y in X, there exist disjoint open sets U and V such that x ∈ U and y ∈ V.
• C) None
• D) For any two distinct points x and y in X, there exists an open set U such that x ∈ U and y ∉ U.