Jointly continuous map

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Suppose are topological spaces. A set map is termed a jointly continuous map if it satisfies the following equivalent conditions:

  1. is a continuous map from equipped with the product topology.
  2. Suppose is a topological space and are all continuous maps. Then, the map given by is a continuous map. (This should be true for every choice of and s).

Relation with separate continuity

Jointly continuous is typically contrasted with the notion of a separately continuous map. A map is separately continuous if it is continuous from the slice topology on . More explicitly, it is continuous in each coordinate fixing the values of all the other coordinates. The slice topology is a finer topology (often, but not necessarily, strictly finer) than the product topology. Thus, joint continuity is a stronger (and in some cases, strictly stronger) condition than separate continuity.

Joint continuity is the correct condition in most circumstances. For instance, if is separately continuous, we cannot be sure whether is continuous, where is the diagonal embedding. More generally, for a separately continuous map, we cannot guarantee continuity under a simultaneous change of both coordinates.

Thus, all definitions that involve continuity from products use joint continuity. Examples include the definition of topological magma, topological monoid, and topological group, the definition of homotopy between maps, and all the definitions/concepts arising from homotopy.