Order and Topology

Question

Given any set, a total order on that set can induce a topology on that same set.
Does the opposite also work ? Given a topology on a set, can it induce an order ( perharps total ) on that set ?

Thanks a lot in advance

Answer 1

Every topology $\tau$ on a set $X$ induces a pre-order $\preceq_\tau$ on $X$: for $x,y\in X$ set $x\preceq_\tau y$ if and only if $x\in\operatorname{cl}_\tau\{y\}$. This is a partial order of $\langle X,\tau\rangle$ is a $T_0$-space. If $X$ is $T_1$, it’s the trivial partial order, equality, and hence isn’t very interesting, so it isn’t much used in general topology; it belongs more to areas of computer science and algebra that use topology. It’s called the specialization pre-order.

Answer 2

Not every topology induces any ordering on a set.

Regard the trivial topology on a set X consisting of only the empty set and X itself. Since this topology will identify the points in X it cannot induce any ordering on X (in sence of any indivual Points).