Mapping Preserves Infima implies Mapping is Continuous in Lower Topological Lattice

Theorem

Let $T = \struct {S, \preceq, \tau}$ and $Q = \struct {X, \preceq', \tau'}$ be complete topological lattices with lower topologies.

Let $f: S \to X$ be a mapping such that

$f$ preserves all infima.

Then $f$ is continuous mapping.

Proof

By assumption:

for all non-empty subsets $Y$ of $S$: $f$ preserves the infimum of $Y$.

Thus by Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice:

$f$ is continuous mapping.

$\blacksquare$

Sources

• Mizar article WAYBEL19:9