Category:Characterization of Compact Element in Complete Lattice

This category contains pages concerning Characterization of Compact Element in Complete Lattice:

Let $L = \struct{S, \preceq}$ be a complete lattice.

Let $a \in S$.

The following statements are equivalent::

$(1)\quad a$ is a compact element

$(2)\quad \forall I \subseteq S : I$ is an ideal $: a \preceq \sup I \implies a \in I$

$(3)\quad \forall A \subseteq S : a \preceq \sup A \implies \exists F \subseteq A : F$ is finite $: a \preceq \sup F$