Coarseness Relation on Topologies is Partial Ordering

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Theorem

Let $X$ be a set.

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

Let $\le$ be the relation on $\mathbb T$:

$\forall \tau_1, \tau_2 \in \mathbb T: \tau_1 \le \tau_2$ iff $\tau_1$ is coarser than $\tau_2$.


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


Proof

It follows directly from the definition of being coarser that:

$\tau_1 \le \tau_2 \iff \tau_1 \subseteq \tau_2$

By Subset Relation is Ordering it thus follows that $\le$ is an ordering.


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

$\blacksquare$


Sources