Non-Empty Compact Closure is Directed
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Theorem
Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice.
Let $x \in S$ such that
- $x^{\mathrm{compact} }$ is a non-empty set,
where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.
Then :$x^{\mathrm{compact} }$ is directed.
Proof
Thus by assumption:
- $x^{\mathrm{compact} }$ is a non-empty set.
Let $y, z \in x^{\mathrm{compact} }$.
By definition of compact closure:
- $y \preceq x$, $z \preceq x$, and $y$ and $z$ are compact.
By definition of compact element:
- $y \ll y$ and $z \ll z$
where $\ll$ denotes the way below relation.
By Way Below is Congruent for Join:
- $y \vee z \ll y \vee z$
By definition:
- $y \vee z$ is a compact element
By definition of supremum:
- $y \vee z \preceq x$
Thus by definition of compact closure:
- $y \vee z \in x^{\mathrm{compact} }$
Thus by Join Succeeds Operands:
- $y \preceq y \vee z$ and $z \preceq y \vee z$
$\blacksquare$
Sources
- Mizar article WAYBEL14:funcreg 1