Saturated set (intersection of open sets)

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 ${\displaystyle S}$  be a subset of a topological space ${\displaystyle X}$ . The saturation ${\displaystyle \operatorname {sat} (S)}$  of ${\displaystyle S}$  is the intersection of all the neighborhoods of ${\displaystyle S}$ .

${\displaystyle \operatorname {sat} (S)=\bigcap {\mathcal {N}}_{S}}$ 

Here ${\displaystyle {\mathcal {N}}_{S}}$  denotes the neighborhood filter of ${\displaystyle S}$ . The neighborhood filter ${\displaystyle {\mathcal {N}}_{S}}$  can be replaced by any local basis of ${\displaystyle S}$ . In particular, ${\displaystyle \operatorname {sat} (S)}$  is the intersection of all open sets containing ${\displaystyle S}$ .

Let ${\displaystyle S}$  be a subset of a topological space ${\displaystyle X}$ . Then the following conditions are equivalent.

• ${\displaystyle S}$  is the intersection of a set of open sets of ${\displaystyle X}$ .
• ${\displaystyle S}$  equals its own saturation.

We say that ${\displaystyle S}$  is saturated if it satisfies the above equivalent conditions. We say that ${\displaystyle S}$  is recurrent if it intersects every non-empty saturated set of ${\displaystyle X}$ .

Properties

Implications

Every G_δ 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 ${\displaystyle X}$ , the following are equivalent.

• Every point ${\displaystyle x\in X}$  has a compact local basis. (This is one of several definitions of locally compact spaces.)
• Every point ${\displaystyle x\in X}$  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.^{[1]: 381, Theorem 2.28} This is a sober variant of the Cantor intersection theorem.

In relation to Baire spaces

For a topological space ${\displaystyle X}$ , the following are equivalent.

• ${\displaystyle X}$  is a Baire space.
• Every recurrent set of ${\displaystyle X}$  is Baire.
• ${\displaystyle X}$  has a Baire recurrent set.

Examples

For a topological space ${\displaystyle X}$ , the following are equivalent.

• Every subset of ${\displaystyle X}$  is saturated.
• The only recurrent set of ${\displaystyle X}$  is ${\displaystyle X}$  itself.
• ${\displaystyle X}$  is a T₁ space.

A subset ${\displaystyle S}$  of a preordered set ${\displaystyle (X,\lesssim )}$  is saturated with respect to the Scott topology if and only if it is upward-closed.^[1]: 380

Let ${\displaystyle (X,\lesssim )}$  be a closed preordered set (one in which every chain has an upper bound). Let ${\displaystyle \max X}$  be the set of maximal elements of ${\displaystyle X}$ . By the Zorn lemma, ${\displaystyle \max X}$  is a recurrent set of ${\displaystyle X}$  with the Scott topology.^{[1]: 397, Proposition 5.6}

References

1. Martin, Keye (1999). "Nonclassical techniques for models of computation" (PDF). Topology Proceedings. 24 (Summer): 375–405. ISSN 0146-4124. MR 1876383. Zbl 1029.06501. Archived (PDF) from the original on 2021-05-10. Retrieved 2022-07-09.

External links

• Saturated set at the nLab