In general topology, a saturated set is a subset of a topological space equal to an intersection of (an arbitrary number of) open sets.
Definition Let be a subset of a topological space . The saturation of is the intersection of all the neighborhoods of .
Here denotes the neighborhood filter of . The neighborhood filter can be replaced by any local basis of . In particular, is the intersection of all open sets containing .
Let be a subset of a topological space . Then the following conditions are equivalent. is the intersection of a set of open sets of . equals its own saturation. We say that is saturated if it satisfies the above equivalent conditions. We say that is recurrent if it intersects every non-empty saturated set of .
Properties Implications Every Gδ set is saturated, obvious by definition. Every recurrent set is dense, also obvious by definition.
In relation to compactness A subset of a topological space is compact if and only if its saturation is compact.
For a topological space , the following are equivalent. Every point has a compact local basis. (This is one of several definitions of locally compact spaces.) Every point has a compact saturated local basis.
In a sober space, the intersection of a downward-directed set of compact saturated sets is again compact and saturated. This is a sober variant of the Cantor intersection theorem.
In relation to Baire spaces For a topological space , the following are equivalent. is a Baire space. Every recurrent set of is Baire. has a Baire recurrent set.
Examples For a topological space , the following are equivalent. Every subset of is saturated. The only recurrent set of is itself. is a T1 space.
A subset of a preordered set is saturated with respect to the Scott topology if and only if it is upward-closed.
Let be a closed preordered set (one in which every chain has an upper bound). Let be the set of maximal elements of . By the Zorn lemma, is a recurrent set of with the Scott topology.
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