Advanced Topology MCQs

By: Prof. Dr. Fazal Rehman Shamil | Last updated: November 25, 2024

  1. Which of the following is true about a continuous function between two topological spaces?
    • (A) It maps open sets to open sets.
    • (B) It maps closed sets to closed sets.
    • (C) It maps compact sets to compact sets.
    • (D) It maps connected sets to connected sets.

    Answer: (C) It maps compact sets to compact sets.

  2. In the context of compactness, which of the following is true about a compact set in a metric space?
    • (A) Every compact set is bounded but not necessarily closed.
    • (B) Every compact set is closed but not necessarily bounded.
    • (C) Every compact set is both bounded and closed.
    • (D) Every compact set is neither bounded nor closed.

    Answer: (C) Every compact set is both bounded and closed.

  3. What is the basis for a topology on a set X?
    • (A) A collection of subsets of X that satisfy certain properties for open sets.
    • (B) A set of functions from X to the real numbers.
    • (C) A set of subsets of X that are closed under unions.
    • (D) A set of subsets of X that are closed under intersections.

    Answer: (A) A collection of subsets of X that satisfy certain properties for open sets.

  4. What is the product topology on a product of two spaces X and Y?
    • (A) The finest topology where all sets are open.
    • (B) The coarsest topology where only the empty set and the whole space are open.
    • (C) The smallest topology for which all projections are continuous.
    • (D) The topology where open sets are products of open sets from X and Y.

    Answer: (C) The smallest topology for which all projections are continuous.

  5. Which of the following spaces is always connected?
    • (A) The empty set.
    • (B) The unit interval [0,1].
    • (C) The set of integers.
    • (D) The discrete topology.

    Answer: (B) The unit interval [0,1].

  6. Which of the following is true about the Cantor set?
    • (A) It is a compact set, but not connected.
    • (B) It is a connected set, but not compact.
    • (C) It is both compact and connected.
    • (D) It is neither compact nor connected.

    Answer: (A) It is a compact set, but not connected.

  7. What does it mean for a space to be Hausdorff (or satisfy the T2 separation axiom)?
    • (A) Any two distinct points have disjoint open neighborhoods.
    • (B) Any two distinct points have overlapping open neighborhoods.
    • (C) Every subset of the space is closed.
    • (D) The space is connected.

    Answer: (A) Any two distinct points have disjoint open neighborhoods.

  8. Which of the following statements is true about the Bolzano-Weierstrass theorem?
    • (A) Every sequence in a compact set has a convergent subsequence.
    • (B) Every sequence in a connected set has a convergent subsequence.
    • (C) Every sequence in a Hausdorff space has a convergent subsequence.
    • (D) Every sequence in a metric space has a convergent subsequence.

    Answer: (A) Every sequence in a compact set has a convergent subsequence.

  9. Which of the following is an example of a non-metric space?
    • (A) The real numbers with the standard topology.
    • (B) The Cantor set.
    • (C) The discrete topology on any set.
    • (D) The Zariski topology on algebraic sets.

    Answer: (D) The Zariski topology on algebraic sets.

  10. What is a homeomorphism between two topological spaces?
  • (A) A continuous function with a continuous inverse.
  • (B) A function that preserves open sets.
  • (C) A continuous function between two spaces that is surjective.
  • (D) A function that is both injective and surjective.