Topology

JKPSC Assistant Professor (Mathematics) · Mathematics · 48 practice questions with answers and explanations: 7 Easy, 35 Medium, 6 Hard.

Sample questions

Sample Easy question 1

The usual topology on ℝ has as a base the:

  1. Half-open intervals [a, b)
  2. Closed intervals
  3. Singletons
  4. Open intervals
Show answer

Answer: D. The half-open intervals give the lower limit topology.

Sample Easy question 2

A set is closed if its complement is:

  1. Open
  2. Empty
  3. Closed
  4. Compact
Show answer

Answer: A. This is the definition of a closed set in a topological space.

Sample Easy question 3

In a discrete topology on a set X:

  1. Open sets are finite
  2. The empty set is open alone
  3. X is open alone
  4. Every subset is open
Show answer

Answer: D. Every singleton is open.

Sample Medium question 4

A continuous real function on a compact metric space:

  1. Is unbounded
  2. Is constant
  3. Attains its maximum and minimum
  4. Need not be bounded
Show answer

Answer: C. This is the extreme value theorem.

Sample Medium question 5

A continuous function on a compact metric space is:

  1. Differentiable
  2. Uniformly continuous
  3. Constant
  4. Not uniformly continuous
Show answer

Answer: B. This is the Heine–Cantor theorem.

Sample Medium question 6

Lebesgue's covering lemma says that every open cover of a compact metric space has a:

  1. Maximum element
  2. Disjoint subcover
  3. Lebesgue number
  4. Countable base
Show answer

Answer: C. Every set of small enough diameter lies in one member of the cover.

Sample Hard question 7

A space is T₁ if and only if every:

  1. Singleton is closed
  2. Singleton is open
  3. Finite set is open
  4. Pair of points has disjoint neighbourhoods
Show answer

Answer: A. A T₂ space is T₁, but not conversely.

Sample Hard question 8

The closure of ℚ in ℝ with the usual topology is:

  1. ℝ − ℚ
  2. ℝ
  3. ℚ
  4. ∅
Show answer

Answer: B. Every real number is a limit of rationals.

Sample Hard question 9

The boundary of the interval (0, 1) in ℝ is:

  1. (0, 1)
  2. ∅
  3. {0, 1}
  4. [0, 1]
Show answer

Answer: C. The boundary is the closure minus the interior.

More Mathematics topics for JKPSC Assistant Professor (Mathematics)

Related articles