| name | geometry-topology |
| description | Formulate and verify geometry and topology arguments. Use for metric and topological spaces, continuity, compactness, connectedness, manifolds, curves and surfaces, homotopy, covering spaces, Euler characteristic, and geometric invariants. |
| license | MIT |
Geometry and Topology
Identify the category
State the objects, maps, equivalence relation, topology or metric, dimension, regularity, orientation, boundary,
and base points. Distinguish local from global claims and intrinsic from embedding-dependent quantities.
Before calculating, name the invariant or obstruction that could decide the problem.
Reason
- For point-set topology, use open-set definitions and verify separation, compactness, connectedness, and continuity hypotheses.
- For manifolds, declare charts and transition regularity; ensure coordinate calculations define global objects.
- For curves and surfaces, distinguish parametrization artifacts from length, curvature, area, and topology.
- For homotopy or covering arguments, track base points, path lifting, induced maps, and group actions.
- Use Euler characteristic, degree, fundamental group, or homology only in the category where it is invariant.
Verify
- Check definitions against canonical examples and edge cases: empty sets, boundaries, noncompact spaces, and singular points.
- Test whether a proposed map is well-defined, continuous, injective, surjective, or a homeomorphism as claimed.
- Verify coordinate overlap and orientation signs; recompute invariants using a different decomposition or chart.
- Search for counterexamples when a hypothesis is dropped and distinguish necessary from sufficient conditions.
- For computation, confirm triangulation independence or convergence and retain exact combinatorial checks.
Deliver
Report the category, hypotheses, invariant or construction, proof, counterexample search, coordinate checks, and
whether the conclusion is local, global, smooth, metric, or purely topological.
Source basis
Original synthesis informed by Hitchman's openly licensed geometry and topology text and Lebl's analysis text;
source details are in ../../docs/TEXTBOOK_SOURCES.md.
