Singleton is Directed and Filtered Subset

Theorem

Let $\struct {S, \precsim}$ be a preordered set.

Let $x$ be an element of $S$.

Then $\set x$ is directed and filtered subset of $S$.

Proof

Let $y, z \in \set x$.

By definition of singleton:

$ y = x \land z = x$

By definition of reflexivity:

$y \precsim x \land z \precsim x$

Thus:

$\exists h \in \set x: y \precsim h \land z \precsim h$

Thus by definition:

$\set x$ is a directed subset of $S$.

$\set x$ is a filtered subset of $S$ by mutatis mutandis.

$\blacksquare$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_0:5