Non-Empty Compact Closure is Directed

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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.


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 $z \preceq y \vee z$