Definition 2

A topological ring is a set A which carries a ring structure and a topology satisfying the following axioms:

(AT$_\mathrm{I}$). The mapping $(x, y) \to x + y$ of $A \times A$ into A is continuous. (AT$_\mathrm{II}$). The mapping $x \to -x$ of A into A is continuous. (AT$_\mathrm{III}$). The mapping $(x, y) \to xy$ of $A \times A$ into A is continuous.

The first two axioms express that the topology of A is compatible with its additive group structure (§ 1, no. 1).

If a ring structure and a topology are given on a set A, they are said to be compatible if they satisfy axioms (AT$_\mathrm{I}$), (AT$_\mathrm{II}$) and (AT$_\mathrm{III}$).