Separately continuous map

Definition

For two spaces

Suppose $A, B, C$ are topological spaces. Suppose $f : A \times B \to C$ is a set map. We say that $f$ is separately continuous if it satisfies the following two conditions:

For every $a \in A$ , the map $b \mapsto f (a, b)$ is a continuous map from $B$ to $C$ .
For every $b \in B$ , the map $a \mapsto f (a, b)$ is a continuous map from $A$ to $C$ .

Equivalently, $f$ is separately continuous if it is continuous as a map from $A \times B$ to $C$ where $A \times B$ is given the slice topology.

For finitely many spaces

Suppose $A_{1}, A_{2}, \dots, A_{n}, C$ are topological spaces. Suppose $f : A_{1} \times A_{2} \times \dots \times A_{n} \to C$ is a set map. We say that $f$ is separately continuous if, for each $i \in {1, 2, \dots, n}$ , and for fixed values of $a_{j}, j \neq i$ the map $a_{i} \mapsto f (a_{1}, a_{2}, \dots, a_{i}, \dots, a_{n})$ is a continuous map from $A_{i}$ to $C$ .

Equivalently, $f$ is separately continuous if it is continuous as a map from $A_{1} \times A_{2} \times \dots \times A_{n}$ , equipped with the slice topology, to $C$ .

Relation with joint continuity

Separately continuous is typically contrasted with the notion of a jointly continuous map. A map $f : A_{1} \times A_{2} \times \dots \times A_{n} \to C$ is jointly continuous if it is continuous from the product topology on $A_{1} \times A_{2} \times \dots A_{n}$ . The product topology is a coarser topology (often, but not necessarily, strictly coarser) than the slice topology. Thus, joint continuity is a stronger (and in some cases, strictly stronger) condition than separate continuity.

Unless otherwise specified, a continuous map from a product space is always taken to be a jointly continuous map, and not merely a separately continuous map.

Joint continuity is the correct condition in most circumstances. For instance, if $f : A \times A \to C$ is separately continuous, we cannot be sure whether $f \circ δ$ is continuous, where $δ : A \to A \times A$ 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.