Image of Compact Subset under Directed Suprema Preserving Closure Operator
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Theorem
Let $L = \struct {S, \preceq}$ be a bounded below algebric lattice.
Let $c: S \to S$ be a closure operator that preserves directed suprema.
Then:
- $c \sqbrk {\map K L} = \map K {\struct {c \sqbrk S, \precsim} }$
where
- $\map K L$ denotes the compact subset of $L$,
- $c \sqbrk S$ denotes the image of $S$ under $c$,
- $\mathord \precsim = \mathord \preceq \cap \paren {c \sqbrk S \times c \sqbrk S}$
Proof
We will prove that:
- $\map K {\struct {c \sqbrk S, \precsim} } \subseteq c \sqbrk {\map K L}$
- $c \sqbrk {\map K L} \subseteq \map K {\struct {c \sqbrk S, \precsim} }$
Thus the result by definition of set equality.
$\blacksquare$
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_8:25