Let $X=...[-2,-1)\cup [0,1)\cup[2,3)\cup...$ and equip $X$ with the two topologies: $\mathcal{T}_1=$ the subspace topology on that $X$ inherits as a subset of the real line, and $\mathcal{T}_2=$ the order topology that $X$ inherits as a subset of the ordered real line.
The question is: are the two spaces $X,\mathcal{T_1}$ and $X,\mathcal{T}_2$ homeomorphic? I know that if $X$ were to be convex, then the subspace and order inherited topologies would be the same, but obviously that is not the case here. My instinct is that they are not:
$[-2,1)$ is open in the subspace topology, $\mathcal{T}_1$, but I don't think that it is open in $\mathcal{T}_2$, and hence there is no homeomorphism which "preserves" open sets between the two spaces. I don't think it's open in $\mathcal{T}_2$, since it we choose a basis element $(a,b)_X$ where $a,b\in X$, then it has the form $(a,b)$ if $|a-b|<1$ or $(a,c)\cup...\cup[d,b)$ (with possibly more elements in the union).
Is my reasoning correct?