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}