Oh good. I know topology better than I thought and didn’t realize it. Mathematical morphology -> complete lattice -> Topology. Does this hold true for all cases of Topology? That’s what I have to find out. “Topology forms Complete Lattice” “The set of all topologies on a set X together with the partial ordering relation ⊆ forms a complete lattice that is also closed under arbitrary intersections. That is, any collection of topologies on X have a meet (or infimum) and a join (or supremum). The meet of a collection of topologies is the intersection of those topologies. The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union. Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.”

Oh good. I know topology better than I thought and didn’t realize it. Mathematical morphology -> complete lattice -> Topology. Does this hold true for all cases of Topology? That’s what I have to find out.
 
“Topology forms Complete Lattice”
 
“The set of all topologies on a set X together with the partial ordering relation ⊆ forms a complete lattice that is also closed under arbitrary intersections. That is, any collection of topologies on X have a meet (or infimum) and a join (or supremum). The meet of a collection of topologies is the intersection of those topologies. The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union.
 
Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.”

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