Connected coverings of the circle are classified by the subgroups $n\mathbb{Z}$ for $n\ge 0$

Given: The quotient circle based at $[0]$.

Based connected coverings correspond to subgroups of the base fundamental group, while unbased connected coverings correspond to conjugacy classes of subgroups (Connected covering spaces are classified by conjugacy classes of fundamental-group subgroups).

The quotient projection $\mathbb{R}\to \mathbb{R}\mathrm{/}\mathbb{Z}$ is a universal covering ($\mathbb{R}\to \mathbb{R}\mathrm{/}\mathbb{Z}$ is a universal covering).

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

Every subgroup of $(\mathbb{Z},+)$ is $n\mathbb{Z}$ for exactly one natural number $n$ (Every subgroup of $(\mathbb{Z},+)$ is $⟨n⟩=n\mathbb{Z}$ for exactly one natural number $n$).

For a covering with nonempty path-connected total space, the number of sheets is the index of its induced subgroup, with both finite or both infinite (For a nonempty path-connected total space, a covering fibre is in bijection with the right cosets of the induced fundamental-group subgroup).

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

The quotient group $(\mathbb{Z},+)\mathrm{/}n\mathbb{Z}$ has the same coset set as $\mathbb{Z}\mathrm{/}n$ (For every $n\in \mathbb{N}$, the congruence-class group $(\mathbb{Z}\mathrm{/}n,+)$ is the quotient group $(\mathbb{Z},+)\mathrm{/}n\mathbb{Z}$).

For $n\ge 1$, the set $\mathbb{Z}\mathrm{/}n$ has exactly $n$ elements; for $n=0$, it is in bijection with $\mathbb{Z}$ (For $n\ge 1$, every class in $\mathbb{Z}\mathrm{/}n$ has one representative $r$ with $0\le r<n$, so $\mid \mathbb{Z}\mathrm{/}n\mid =n$; while $\mathbb{Z}\mathrm{/}0$ is in bijection with $\mathbb{Z}$).

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

The quotient map is open, and every real interval of length below one maps homeomorphically to its image in the quotient circle (The quotient map is open, and every interval shorter than one embeds in $\mathbb{R}\mathrm{/}\mathbb{Z}$).

Every nonempty convex interval is simply connected (Every nonempty convex subset of ${\mathbb{R}}^n$ is simply connected).

Local path-connectedness requires arbitrarily small open path-connected neighbourhoods, while semilocal simple connectedness requires a neighbourhood whose inclusion induces the trivial fundamental-group map (Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point, Semilocally simply connected spaces with explicit basepoint convention).

A pointed homeomorphism induces a fundamental-group isomorphism (Induced fundamental-group maps are well defined, functorial and invariant under based homotopy).

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).