Manifoldet

Manifoldet is a term found in some speculative and pedagogical writings to denote a generalized, manifold-like object that foregrounds a layered local structure. The name combines manifold with a diminutive suffix, signaling a small or elementary extension of the classical notion. In these discussions, a manifoldet is intended to behave like a manifold on typical points while allowing a discrete or stratified layer of data to vary from point to point, offering a way to discuss mild singularities or discrete labels without abandoning smooth local geometry.

A working informal definition views a manifoldet of dimension n as a second-countable, Hausdorff space M equipped with an atlas of charts that locally model neighborhoods as a product of an open set of R^n with a finite discrete label set. In more structured phrasing, there is a locally finite collection of charts φ_i: U_i → V_i ⊂ R^n × F_i, where F_i is finite, and the transition maps are smooth in the R^n coordinates and compatible with the discrete labels. Equivalently, one can think of M as having a locally constant labeling that induces a stratification by depth, in which each stratum is an n-dimensional smooth manifold and the strata fit together in a controlled way.

Relation to existing concepts is central to its role in discussions. If the discrete label is constant across M, a manifoldet reduces to a standard manifold. If the labeling yields distinct strata, the space resembles a stratified or layered manifold-like object, useful for illustrating how discrete data can be layered over a smooth base.

Examples commonly cited include a disjoint union of two copies of R^n, viewed as a two-sheeted manifoldet, and more elaborate stratifications where a smooth region coexists with labeled discrete features. The concept is primarily of interest as a teaching or exploratory device rather than as a widely adopted mathematical object. See also: manifold, stratified space, fiber bundle.