Specialization (pre)order - Upper and Lower Sets

Upper and Lower Sets

Let X be a topological space and let ≤ be the specialization preorder on X. Every open set is an upper set with respect to ≤ and every closed set is a lower set. The converses are not generally true. In fact, a topological space is an Alexandrov space if and only if every upper set is open (or every lower set is closed).

Let A be a subset of X. The smallest upper set containing A is denoted ↑A and the smallest lower set containing A is denoted ↓A. In case A = {x} is a singleton one uses the notation ↑x and ↓x. For x ∈ X one has:

• ↑x = {y ∈ X : x ≤ y} = ∩{open sets containing x}.
• ↓x = {y ∈ X : y ≤ x} = ∩{closed sets containing x} = cl{x}.

The lower set ↓x is always closed; however, the upper set ↑x need not be open or closed. The closed points of a topological space X are precisely the minimal elements of X with respect to ≤.