Coarseness Relation on Topologies is Partial Ordering

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Let $S$ be a set.

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

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

$\forall \tau_1, \tau_2 \in \mathbb T: \tau_1 \le \tau_2$ if and only if $\tau_1$ is coarser than $\tau_2$.

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


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 are not necessarily Comparable by Coarseness, it follows that such an ordering is not always total.