topological isomorphism between a topological group and the identity component of a topological group

Question

Let $\Bbb R$ be the group of real numbers with the usual topology and $\Bbb Z$ the group of integers with the discrete topology. Is $\Bbb R$ topological isomorphism by the identity component of $(\Bbb R\bigoplus \Bbb Z)/H$ where $H$ is the subgroup $\{(n,-2n);n\in \Bbb Z\}$?

Answer 1

Hint:

Let $\Bbb Z_2$ denote the cyclic group of order 2, with the discrete topology.

Define a map $\phi: \Bbb R \oplus \Bbb Z \to \Bbb R \oplus \Bbb Z_2$ as $(x,n) \mapsto \big(x+\frac{1}{2}n, \;n \pmod 2\big)$.

Then $\phi$ is a surjective homomorphism with kernel $H$, so what does the first isomorphism theorem tell you?