Definition:Principal Open Subset of Spectrum

Definition
Let $A$ be a commutative ring with unity.

Let $f \in A$.

The principal open subset determined by $f$ of the spectrum $\Spec A$ is the complement of the vanishing set $\map V f$:
 * $\map D f = \Spec A - \map V f$

That is, it is the set of prime ideals $\mathfrak p \subseteq A$ with $f \notin \mathfrak p$.

Also denoted as
The principal open subset is also denoted $\map X f$ or $X_f$.

Also known as
A principal open subset is also known as a basic open set.

Also see

 * Definition:Zariski Topology on Spectrum of Ring