nonproper

Nonproper is an adjective used to indicate that something does not meet the standard of propriety or the formal property of being proper in a given domain. In general English, it can describe behavior or actions that are not appropriate or acceptable. The term is more common in technical disciplines where properness has a precise meaning.

In mathematics, a map is called proper if the preimage of every compact subset of the codomain is compact in the domain. Nonproper maps are those that fail this condition; equivalently, there exists a compact subset of the codomain whose preimage is not compact. A standard example is the projection from the plane to a coordinate line, p: R^2 -> R, p(x,y)=x. The set [a,b] in R is compact, but p^{-1}([a,b]) = [a,b] x R is not compact, so p is nonproper. Nonproperness can affect how a map interacts with limits, compactness arguments, and fiber behavior.

Outside topology, the term appears as a direct negation of proper in definitions or theorems, and may be encountered in algebraic geometry, complex analysis, or dynamical systems, among others, whenever properness is a stated hypothesis. Hyphenation varies by style guide; some writers use non-proper, others prefer nonproper.

See also: proper, proper map, non-proper morphism, improper.