A topological ring is a ring $R$ which is also a topological space such that both the addition and the multiplication are continuous as maps.
$F$ is a topological field, if $F$ is a topological ring, and the inversion operation is continuous, when restricted to $F\backslash\{0\}$.
If $F$ is a path-connected field, then $F$ must be topologically isomophic to the real number field $\mathbb{R}$ or the complex number field $\mathbb{C}$?