Define distance between points.
A metric determines balls, neighborhoods and quantitative notions of closeness.
Side 136
Geometry & Topology
Two ways to study space: geometry asks about measurement and curvature; topology asks what survives continuous deformation when exact lengths and angles are ignored.
space→shape→metric→continuity→invariant
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Geometry studies structure with measurement.
Distance, angle and curvature distinguish shapes that topology may treat as equivalent.
Find locally shortest or extremal paths.
On curved spaces, geodesics generalize straight lines.
Measure deviation from flatness.
Curvature can be local while constraining global geometry.
Identify transformations preserving geometric structure.
Rigid motions and broader symmetry groups classify spatial regularity.
Topology keeps continuity and discards exact measurement.
Topological equivalence depends on continuous deformation rather than matching coordinates or distances.
Define local neighborhoods abstractly.
Open sets generate the topology and formalize continuity without requiring a metric.
Preserve nearby structure.
Continuity prevents tearing or jumping while allowing stretching and bending.
Match spaces continuously in both directions.
Homeomorphic spaces are topologically the same despite geometric differences.
Track properties preserved by deformation.
Connectedness, holes and compactness help distinguish topological types.
Surfaces reveal the interaction between local and global structure.
A surface can look locally simple while having nontrivial global topology.
Model a space that locally resembles Euclidean space.
Manifolds support calculus while allowing global curvature and topology.
Ask whether a consistent orientation exists.
The Möbius strip is the classic example of a non-orientable surface.
Count handle-like features.
Genus helps classify closed orientable surfaces.
Distinguish interior from edge.
Adding or removing boundaries changes both geometry and topology.
Geometry and topology constrain one another.
Global topological facts can limit possible geometric structures and vice versa.
Combine vertices, edges and faces into an invariant.
It remains stable under many decompositions and helps classify surfaces.
Place one space inside another.
Embedding problems reveal how intrinsic structure interacts with ambient space.
Classify maps up to continuous deformation.
Homotopy tracks deeper structure than pointwise equality.
Use geometric tools to study topological spaces.
Modern problems often require both measurement and deformation-invariant reasoning.
Geometry measures; topology preserves. The same object can be geometrically different yet topologically equivalent, depending on which properties the question treats as essential.