Semicompleteness

Semicompleteness is a term used in several branches of mathematics to denote a property that is weaker than full completeness, yet strong enough to guarantee meaningful limiting behavior in many contexts. Because there is no universal definition, the precise meaning of semicompleteness is defined within each theory or framework in which it appears.

In dynamical systems and the theory of differential equations on manifolds, semicompleteness often refers to a condition on flows generated by vector fields. Roughly, a semicomplete vector field is one for which the flow exists and can be extended in at least one time direction for every initial point, though not necessarily for all complex times or for all time values. This property is weaker than completeness of the flow.

In metric and topological settings, semicompleteness appears as a property of sequences, nets, or Cauchy-type convergence relative to a chosen framework. For example, a space may be called semicomplete if every Cauchy sequence that lies in a specific substructure has its limit again inside that substructure, or if a particular completion or compactification retains the substructure without adding new limit points. The exact formulation depends on the chosen structure and the intended applications.

Relation to other notions: complete spaces satisfy semicompleteness in the usual sense, but semicompleteness does not imply full completeness. In practice, the term is used more as a heuristic to indicate a usable, though weaker, form of convergence or extendability.

See also: completeness, dynamical systems, flows, vector fields, compactness.