Filtered iff Upper Closure Filtered

Theorem

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

Let $H$ be a non-empty subset of $S$.

Then $H$ is filtered if and only if $H^\succsim$ is filtered

where $H^\succsim$ denotes the upper closure of $H$.

Proof

This follows by mutatis mutandis of the proof of Directed iff Lower Closure Directed.

$\blacksquare$

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_0:35