An irrational line as a dense immersed Lie subgroup of a torus

By [A1], $i$ is an injective immersed homomorphism with dense image. Since its kernel is trivial, the canonical image structure in [A2] is transported from the one-dimensional source $\mathbb{R}$.

The image is proper. The point $(1,e^{\pi i\alpha })$ is not in it: equality of the first coordinate would force $t=n\in \mathbb{Z}$, while equality of the second would make $\alpha (n-\frac{1}{2})$ an integer, impossible because a nonzero rational multiple of irrational $\alpha$ is irrational. A proper dense subset is not closed.

For each $j\ge 1$, let $q_j$ be the least positive integer satisfying $\parallel q_j\alpha \parallel <1\mathrm{/}j$, whose existence is the finite-pigeonhole calculation in [A1]. Irrationality makes every fixed $\parallel q\alpha \parallel$ positive, so $q_j\to \mathrm{\infty }$. Nevertheless $i(q_j)=(1,e^{2\pi i\alpha q_j})\to (1,1)=i(0)$ in the ambient subspace. Hence the inverse of $i$ on its image is not continuous, so [F1] shows that the subgroup is not embedded. Leastness makes the sequence choice-free; ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited only through the general image supplier [A2]. The source dimension is exactly one, its tangent $(1,\alpha )$ is nonzero, and no endpoint is present.