Theoremina

Theoremina is a fictional mathematical construct used in introductory texts and online learning platforms to illustrate local-to-global gluing principles.

In the standard fictional formulation, let X be a connected topological space with an open cover {Ui}, and suppose each Ui carries a local structure si (for example a geometric embedding or a bundle trivialization). If on all overlaps Ui ∩ Uj the local structures agree and satisfy a coherence condition (the gluing or cocycle condition), then there exists a global structure S on X such that S restricted to Ui equals si for all i.

A simple example in geometry is that two local charts with compatible transitions on overlaps determine a global manifold structure; in tiling theory, local matching rules imply a global tiling when coherence holds. The concept is also used to illustrate how local constraints can, under suitable conditions, be assembled into a consistent global object.

Relation to real mathematics is that sheaf theory, descent, and related frameworks formalize these ideas, providing precise criteria for when local data glue to global structures. Theoremina mirrors these ideas as an educational abstraction rather than an actual proven result, highlighting the distinction between local data and global existence.

See also: local-to-global principles, cocycle condition, gluing lemma, sheaf theory, descent.