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

$\mathrm{sat}(S)=\bigcap {𝒩}_S$

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

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

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

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

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 $X$, the following are equivalent.

• Every point $x\in X$ has a compact local basis. (This is one of several definitions of locally compact spaces.)
• Every point $x\in X$ 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. This is a sober variant of the Cantor intersection theorem.

In relation to Baire spaces

For a topological space $X$, the following are equivalent.

• $X$ is a Baire space.
• Every recurrent set of $X$ is Baire.
• $X$ has a Baire recurrent set.

Examples

For a topological space $X$, the following are equivalent.

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

A subset $S$ of a preordered set $(X,\lesssim )$ is saturated with respect to the Scott topology if and only if it is upward-closed.

Let $(X,\lesssim )$ be a closed preordered set (one in which every chain has an upper bound). Let $\max X$ be the set of maximal elements of $X$. By the Zorn lemma, $\max X$ is a recurrent set of $X$ with the Scott topology.

External links

• Saturated set at the nLab