Definition:Filtered Subset

Definition

Let $\left({S, \precsim}\right)$ be a preordered set.

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

Then $H$ is a filtered subset of $S$ if and only if:

$\forall x, y \in H: \exists z \in H: z \precsim x$ and $z \precsim y$

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:def 2