Every connected covering of the circle is regular

Given: A connected covering $p:(E,e_0)\to (\mathbb{R}\mathrm{/}\mathbb{Z},[0])$.

For a covering with path-connected total space and path-connected locally path-connected base, regularity is equivalent to normality of the induced subgroup in the base fundamental group (A connected covering is regular exactly when its induced subgroup is normal, exactly when deck transformations act transitively on a fibre).

Degree gives an isomorphism from the circle fundamental group to $(\mathbb{Z},+)$ ($\mathrm{Deg}:{\pi }_1(\mathbb{R}\mathrm{/}\mathbb{Z},[0])\to (\mathbb{Z},+)$ is an isomorphism).

Every subgroup of an abelian group is normal (Every subgroup of an abelian group is normal).

The additive group of $\mathbb{Z}$ is abelian (The integers form a commutative ring).

The quotient circle is path-connected ($\mathbb{R}\mathrm{/}\mathbb{Z}$ is compact and path-connected).

Open quotient arcs are homeomorphic to convex real intervals and form arbitrarily small path-connected neighbourhoods of circle points, so the quotient circle is locally path-connected (The quotient map is open, and every interval shorter than one embeds in $\mathbb{R}\mathrm{/}\mathbb{Z}$, Every nonempty convex subset of ${\mathbb{R}}^n$ is simply connected, Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point).

Local path-connectedness lifts from the base of a covering to its total space (Local path-connectedness lifts and descends along covering maps).

A connected locally path-connected space is path-connected (A connected, locally path-connected space is path-connected, because its path components are open).