Coarseness Relation on Topologies is Partial Ordering

Theorem
Let $$X$$ be a set.

Let $$\mathbb T$$ be the set of all topologies on $$X$$

Let $$\le$$ be the relation on $$\mathbb T$$:
 * $$\forall \vartheta_1, \vartheta_2 \in \mathbb T: \vartheta_1 \le \vartheta_2 \ \stackrel {\mathbf {def}} {=\!=} \ \vartheta_1$$ is coarser than $$\vartheta_2$$.

Then $$\le$$ is a partial ordering on $$\mathbb T$$.

Proof
Follows directly from the definition that:
 * $$\vartheta_1 \le \vartheta_2 \iff \vartheta_1 \subseteq \vartheta_2$$

We have that the Subset Relation is Ordering and so $$\le$$ is also an ordering.

From Topologies Not Always Comparable by Coarseness, it follows that such an ordering is not always total.