Another example of a preorder is the [Specialization preorder](https://en.wikipedia.org/wiki/Specialization_(pre)order) for a topology \$$\tau\$$. This is defined:

$$x \leq y \Longleftrightarrow x \in \mathbf{cl}(\\{y \\})$$

Where \$$\mathbf{cl}(\cdot)\$$ is the [closure](https://en.wikipedia.org/wiki/Closure_(topology)) operator for \$$\tau\$$.

As Daniel Michael Cicala mentioned:

> Here are two more examples of preorders. Take your favorite set \$$X\$$.

> 1) For all \$$x\$$ in \$$X\$$, setting \$$x\leq x\$$ gives you a preorder.

> 2) For all \$$x, y\$$ in \$$X\$$, setting \$$x\leq y\$$ gives you a preorder.

(1) is the specialization preorder of the [discrete topology](https://en.wikipedia.org/wiki/Discrete_space) over \$$X\$$.

(2) is the specialization preorder of the [trivial topology](https://en.wikipedia.org/wiki/Trivial_topology) over \$$X\$$.