Compact Closure is Subset of Way Below Closure
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Theorem
Let $L = \struct {S, \preceq}$ be an ordered set.
Let $x \in S$.
Then $x^{\mathrm {compact} } \subseteq x^\ll$
where
- $x^{\mathrm {compact} }$ denotes the compact closure of $x$,
- $x^\ll$ denotes the way below closure of $x$.
Proof
Let $y \in x^{\mathrm {compact} }$.
By definition of compact closure:
- $y \preceq x$ and $y$ is compact.
By definition of compact:
- $y \ll y$
where $\ll$ denotes the way below relation.
By Preceding and Way Below implies Way Below and definition of reflexivity:
- $y \ll x$
Thus by definition of way below closure:
- $y \in x^\ll$
$\blacksquare$
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_8:6