Lower Closure of Element is Closed under Directed Suprema
Jump to navigation
Jump to search
Theorem
Let $L = \struct {S, \preceq}$ be an up-complete ordered set.
Let $x \in S$.
Then $x^\preceq$ is closed under directed suprema,
where $x^\preceq$ denotes the lower closure of $x$.
Proof
Let $D$ be a directed subset of $S$ such that
- $D \subseteq x^\preceq$
By Lower Closure of Element is Ideal:
- $x^\preceq$ is directed.
By definition of up-complete:
- $D$ and $x^\preceq$ admit suprema.
- $\sup D \preceq \map \sup {x^\preceq}$
By Supremum of Lower Closure of Element:
- $\sup D \preceq x$
Thus by definition of lower closure of element:
- $\sup D \in x^\preceq$
$\blacksquare$
Sources
- Mizar article WAYBEL11:8