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A Hausdorff, second-countable, locally Euclidean space.
The manifold condition asserts existence of local Euclidean charts; a particular topological atlas is not additional structure. This atlas adopts the Hausdorff, second-countable convention.
wp-manifoldnlab-topological-manifoldtopology-without-tearsEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Impose all manifold conditions on one topology.
Feed several structures into a construction junction and impose compatibility between them.
wp-manifoldtopology-without-tearsEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Choose complex charts with holomorphic transition maps.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-complex-manifoldlee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Choose a maximal atlas whose transition functions are smooth.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
nlab-smooth-manifoldlee-smooth-manifolds