reguljär

Reguljär is a Swedish adjective borrowed from Latin regulāris and French régulier. In mathematics and closely related fields it is used to describe something that is regular in a well-defined sense: non-singular, smooth or well-behaved within a given context. The exact meaning depends on the branch of mathematics being considered.

In topology, reguljär rymd (regular space) refers to a space that satisfies a separation axiom ensuring the distinction between points and closed sets. Formally, a topological space X is regular if, for every point x and closed set F not containing x, there exist disjoint open sets U containing x and V containing F. Regular spaces are closely related to the T3 separation axiom.

In differential topology and geometry the term appears in several closely related notions. A differentiable map f: M -> N has a reguljär värde y if, for every x in the preimage f^{-1}(y), the derivative Df_x is surjective. The preimage of a reguljär värde is a smooth submanifold. A reguljär kurva (regular curve) is a parametrized curve with non-vanishing derivative, i.e., its velocity vector never vanishes.

In algebraic geometry and the theory of schemes, a point on a variety or scheme is reguljär if the local ring is regular. This is equivalent to the tangent space having the expected dimension; a reguljär punkt is non-singular. Regularity in this sense often aligns with geometric intuition of smoothness.

Because reguljär is context dependent, it is best understood from the specific field’s definition. The term is commonly contrasted with singular or irregular, to indicate a breakdown of smoothness, well-definedness, or expected dimension.