Is it appropriate to ask the community to check my proof? I am rereading Munkres Topology and trying to do the HW. This is my attempt for #10 on page 101.
Show that every order topology is Hausdorff.
Proof: Suppose that $x_1, x_2$ are elements of $X$, and $x_1 < x_2$.
Suppose that $x_2$ is the next element after $x_1$, and suppose that $$x_0 < x_1 < x_2 < x_3.$$ Then $x_1$ is an element of $(x_0, x_2) = U_1$. Then $x_2$ is an element of $(x_1, x_3) = U_2$. $U_1$ and $U_2$ are disjoint.
Case 2: Suppose that there might not be "next elements" in $X$ but $x_1 < x_2$. Then there exists $$a < x_1 < b < x_2 < c.$$ Then $x_1$ is an element of $(a, b) = U_1$. Then $x_2$ is an element of $(b, c) = U_2$. $U_1$ and $U_2$ are disjoint.
Thus $X$ is Hausdorff.
Is this on the right track?