Complete variety

In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism

is a closed map, i.e. maps closed sets onto closed sets. This can be seen as an analogue of compactness in algebraic geometry: a topological space X is compact if and only if projection along X is a closed map in any product X × Y, equipped with product topology.