Topology
38 questions across 2 topics. Also examined in CNET.
Separation Axioms, Connectedness and Compactness
- 52Topology — a two-valued map and connectedness: only one direction holdsMSET June 2025Separation Axioms, Connectedness and CompactnessEasy
- 53Topology — compactness in a discrete spaceMSET June 2025Separation Axioms, Connectedness and CompactnessEasy
- 55Topology — products and punctured circlesMSET June 2025Separation Axioms, Connectedness and CompactnessMedium
- 56Topology — normality via shrinking neighbourhoodsMSET June 2025Separation Axioms, Connectedness and CompactnessMedium
- 57Topology — the curve x^2y^2=1MSET June 2025Separation Axioms, Connectedness and CompactnessEasy
- 3Topology — boundedness of continuous functions characterises compactnessMSET April 2024Separation Axioms, Connectedness and CompactnessHard
- 51A homeomorphic image of a disconnected unionMSET April 2024Separation Axioms, Connectedness and CompactnessHard
- 55Connectedness — what a shared point buys you (and what it doesn't)MSET April 2024Separation Axioms, Connectedness and CompactnessHard
- 58The union of the two coordinate axes is connectedMSET April 2024Separation Axioms, Connectedness and CompactnessMedium
- 60Half-open squares — when a reflection gives a homeomorphism, and when nothing doesMSET April 2024Separation Axioms, Connectedness and CompactnessHard
- 4The agreement set of two continuous maps is always closedMSET March 2023Separation Axioms, Connectedness and CompactnessEasy
- 60A chain of overlapping unit circles — connected, but unboundedMSET September 2021Separation Axioms, Connectedness and CompactnessMedium
- 15Every set is clopen and bounded in the discrete metric — only infinite sets fail to be compactMSET December 2020Separation Axioms, Connectedness and CompactnessMedium
- 60Two components of an open disc each, regardless of gap size or how the split arises, are all homeomorphicMSET December 2020Separation Axioms, Connectedness and CompactnessHard
- 60Two simple closed curves are homeomorphic to each other, but not to a branched "X"MSET June 2019Separation Axioms, Connectedness and CompactnessMedium
- 20The topologist's sine curve: connected but not path connectedMSET January 2018 Paper IIISeparation Axioms, Connectedness and CompactnessMedium
- 37An indiscrete space with two points cannot separate themMSET January 2018 Paper IIISeparation Axioms, Connectedness and CompactnessEasy
- 39Local connectedness means components of open sets are openMSET January 2018 Paper IIISeparation Axioms, Connectedness and CompactnessMedium
- 22Path connected Rightarrow connected; the converses failMSET April 2017 Paper IIISeparation Axioms, Connectedness and CompactnessEasy
- 37In a Hausdorff space every option holds for a limit pointMSET April 2017 Paper IIISeparation Axioms, Connectedness and CompactnessMedium
- 39The irrationals: every compact subset has empty interiorMSET April 2017 Paper IIISeparation Axioms, Connectedness and CompactnessMedium
Basis, Dense Sets, Subspace and Product Topology
- 50Topology — (-1,1)congRcong a parabolaMSET June 2025Basis, Dense Sets, Subspace and Product TopologyEasy
- 58Topology — comparing metric topologiesMSET June 2025Basis, Dense Sets, Subspace and Product TopologyMedium
- 59Topology — the upper-limit topologyMSET June 2025Basis, Dense Sets, Subspace and Product TopologyMedium
- 53The precise definition of a regular spaceMSET April 2024Basis, Dense Sets, Subspace and Product TopologyHard
- 57Subspace topology — what transfers cleanly, and what doesn'tMSET April 2024Basis, Dense Sets, Subspace and Product TopologyHard
- 51Why R and R^2 cannot be homeomorphicMSET March 2023Basis, Dense Sets, Subspace and Product TopologyHard
- 54Density is preserved by continuous surjectionsMSET March 2023Basis, Dense Sets, Subspace and Product TopologyEasy
- 55Spotting the collection that can't isolate a single integer sliceMSET March 2023Basis, Dense Sets, Subspace and Product TopologyHard
- 56A K-topology-style refinement — Hausdorff, but not regularMSET March 2023Basis, Dense Sets, Subspace and Product TopologyMedium
- 87The variational form of Poisson's equationMSET March 2023Basis, Dense Sets, Subspace and Product TopologyMedium
- 13An open interval can be mapped continuously onto a closed one, never the reverseMSET December 2020Basis, Dense Sets, Subspace and Product TopologyMedium
- 59Only the closure containment survives for a general continuous map; interior and boundary containments can failMSET December 2020Basis, Dense Sets, Subspace and Product TopologyHard
- 100The limit points of a dense set fill up the whole lineMSET December 2020Basis, Dense Sets, Subspace and Product TopologyMedium
- 59Closure in the upper-ray topology: everything not strictly aboveMSET June 2019Basis, Dense Sets, Subspace and Product TopologyEasy
- 38Boundedness is not topological: arctan makes R boundedMSET January 2018 Paper IIIBasis, Dense Sets, Subspace and Product TopologyMedium
- 36"Open in Y" does not transfer to X unless Y is openMSET April 2017 Paper IIIBasis, Dense Sets, Subspace and Product TopologyEasy
- 38An open surjection is a quotient map only if it is also continuousMSET April 2017 Paper IIIBasis, Dense Sets, Subspace and Product TopologyMedium