Topology MCQ Set - Module I

Which of the following is the correct definition of a topology on a set X?

• A collection of subsets of X that contains ∅ and X

• A collection of subsets of X that is closed under arbitrary unions and finite intersections, and contains ∅ and X

• A collection of all possible subsets of X

• A collection of subsets of X that is closed under finite unions only

Let X = {a, b, c}. Which of the following is NOT a topology on X?

• τ = {∅, X, {a}, {b, c}}

• τ = {∅, X, {a}, {a, b}, {a, c}}

• τ = {∅, X}

• τ = {∅, X, {a}, {b}, {a, b}, {b, c}, {c}}

The standard topology on ℝ (real numbers) is generated by:

• All finite sets

• All countable sets

• All open intervals (a, b) where a < b

• All closed intervals [a, b] where a ≤ b

In a metric space (X, d), the metric topology is defined by:

• Open balls B(x, r) = {y ∈ X : d(x, y) < r}

• Closed balls B[x, r] = {y ∈ X : d(x, y) ≤ r}

• Single point sets {x}

• Complements of finite sets

A neighborhood of a point x in a topological space (X, τ) is:

• Any open set containing x

• Any set containing x

• Any set N such that there exists an open set U with x ∈ U ⊆ N

• The smallest open set containing x

The interior of a set A in a topological space is:

• The smallest closed set containing A

• The largest open set contained in A

• The set of all limit points of A

• The complement of the closure of A

If A = [0, 1] in ℝ with the usual topology, then the interior of A is:

• [0, 1]

• (0, 1)

• {0, 1}

• ∅

The exterior of a set A is defined as:

• The complement of A

• The closure of the complement of A

• The interior of the complement of A

• The boundary of A

A point x is a boundary point of set A if:

• x ∈ A

• x ∉ A

• Every neighborhood of x contains points in both A and its complement

• x is an interior point of A

In ℝ with usual topology, the boundary of the set (0, 1) ∪ (1, 2) is:

• {0, 1, 2}

• {0, 2}

• {1}

• [0, 2]