is the torus homeomorphic to a piece of cylinder surface in the quotient?

Question

Considering over $X=S^1\times\mathbb R$, with the usual induced topology and the equivalence relation $(u,t )R(u',t')$ if and only if $u'=\pm u, t-t' \in \mathbb Z$.

Tell if $(X/R,\epsilon_2/R)$ is T2, connected compact and homeomorphic to the torus $S^1\times S^1$

any idea

Answer 1

Being the first space $S^1$, the quotient defined by $R$ makes it $\mathbb{RP1}$ which is isomorphic to $S^1$ itself. In the second coordinate you have the exact definition of $S^1$ so your quotient space is actually isomorphic to a torus. Everything else can be answered from here.