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 $\mathbb T$:

$\forall \vartheta_1, \vartheta_2 \in \mathbb T: \vartheta_1 \le \vartheta_2 := \vartheta_1$ 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.

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$