{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

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

• A) False
• B) True
• C)
• D)

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

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

A topological space is a \(T_{1}\)- space if and only if each of its finite subset is ....

• A) an open set
• B) neither open nor closed.
• C) both open and closed.
• D) a closed set.

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

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

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

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

A topological space is a T_{1}- space if and only if each of its finite subset is ....

• A) both open and closed.
• B) neither open nor closed.
• C) a closed set.
• D) an open set

The ________ command calculates the eigen value of a ________ matrix.

• A) eigen, square
• B) value, rectangular
• C) eig, square
• D) eig, rectangular

What MATLAB function is used to find the inverse of a matrix?

• A) eig()
• B) eye()
• C) inv()
• D) inverse()

If we use [P, D] = eig(B), P determines ______ and D determines _________.

• A) square matrix, rectangular matrix
• B) rectangular matrix, square matrix
• C) eigen values, eigen vectors
• D) eigen vectors, eigen values

________ command can be used for the solution of system of linear equations.

• A) xlim
• B) label
• C) eig
• D) rref

_________ command calculates the inverse of the matrix A.

• A) eig(A)
• B) none of these
• C) inv(A)
• D) sym(A)

Using MATLAB, solution of system of linear equations is obtained by _____________ command .

• A) solve
• B) sym
• C) inv
• D) eig

To calculate the reduced echelon form of a matrix E=[0 1 3 4; 2 4 1 5; 6 1 8 6], we will write __________ in MATLAB.

• A) rref
• B) eig(E)
• C) eig
• D) rref(E)