Definition:Topological Field

Definition
Let $\left({F, +, \circ}\right)$ be a field with zero $0_F$.

Let $\tau$ be a topology on $F$.

Let the mapping $\phi: F \setminus \left\{{0_F}\right\} \to F$ be defined as:
 * $\phi \left({x}\right) = x^{-1}$ for each $x \in F \setminus \left\{{0_F}\right\}$

Then $\left({F,+,\circ,\tau}\right)$ is a topological field iff:


 * $(1): \quad \left({F, +, \circ, \tau}\right)$ is a topological ring


 * $(2): \quad \phi$ is a $\tau'$-$\tau$-continuous mapping, where $\tau'$ is the $\tau$-relative subspace topology on $F\setminus\left\{{0_F}\right\}$.