Continuous iff Directed Suprema Preserving
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Theorem
Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be complete topological lattices with Scott topologies.
Let $f: S \to T$ be a mapping.
Then $f$ is continuous if and only if $f$ preserves directed suprema.
Proof
Sufficient Condition
Assume that
- $f$ is continuous.
- for all directed set $\struct {D, \precsim}$ and net $N:D \to S$ in $S$: $\map f {\liminf N} \preceq_2 \map \liminf {f \circ N}$
$\Box$
Necessary Condition
Thus by Mapping Preserves Directed Suprema implies Mapping is Continuous:
- if $f$ preserves directed suprema, then $f$ is continuous.
$\blacksquare$
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 WAYBEL17:22