- 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.
- 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.
- 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.
- 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.
- 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].
- 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.
- 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.
- 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.
- 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.
- 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.