Which of the following statements is true about metric spaces?

• A) Every metric space is not Hausdorff.
• B) Every metric space is complete.
• C) Every metric space is Hausdorff.
• D) Every metric space is discrete.

If $\left( {X,\tau } \right)$ be a separable topology then it must have countable dense set.

• A) True
• B) False
• C)
• D)

Which of the following is a correct reason why every subspace of a Hausdorff space is also Hausdorff?

• A) Because the subspace have different separation properties from the parent space.
• B) Because every subspace is closed in a Hausdorff space.
• C) Because every subspace of a topological space is always Hausdorff.
• D) Because the subspace inherits the separation properties from the parent space.

Which of the following statements about an indiscrete topological space is true?

• A) It is always a $T_{0}$ space.
• B) It is never a $T_{0}$ space.
• C) It may be a $T_{0}$ space.
• D) None of these

Which of the following is not an example of compact spaces?

• A) A finite set with any toplogy.
• B) An infinite set with discrete topology.
• C) A set with topology containing finite number of elements.
• D) A set with indiscrete toplogy.

Which of the following statements correctly compares T1​ and T0 properties?

• A) T1​ property implies T0 property, but not vice versa.
• B) T0​ property is a stronger property than T1​ property.
• C) T0​ property implies T1​ property, but not vice versa.
• D) T1​ property is equivalent to T0 property.

Consider a set $X$ with the indiscrete topology. What can be said about the singleton sets in $X$?

• A) They are neither open nor closed.
• B) They are closed but not open.
• C) They are open but not closed.
• D) They are open and closed.

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

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

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

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

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

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

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

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) None of them.
• B) It is Haussdorff.
• C) It is metrizable.
• 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) None of them
• B) \left\{ {\{ 1\} ,\{ 2\} ,X} \right\}
• C) \left\{ {\{ 1\} ,\{ 1,2\} ,X} \right\}
• D) \left\{ {\{ 1\} ,\{ 2\} ,\{ 1,2\} ,X} \right\}

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

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

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

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

LetX = 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) It must be an open set.
• C) Neither of its point is an interior point.
• 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) Every element of X has uncountable local base.
• B) \left( {X,\tau } \right) be a topological space.
• C) The local base of the element 4 is \emptyset
• D) \left( {X,\tau } \right) be a first countable space.

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

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

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

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

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