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\\).

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

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\\).