The basic property of scheme

Question

Suppose $X$ is a scheme and $x$ is a point in $X$. Then there exists an open affine subscheme of $X$ named $SpecA$ such that $x\in SpecA$. I want to know if $\overline{\{x\}} \subseteq SpecA $?

Answer 1

No. If a scheme $S$ is irreducible, then there exists a point $x \in S$ such that $\overline{\{x\}} = S$, so if the scheme is not affine, your property cannot hold. For example, consider a projective space $\mathbb{P}^n_k$ over a field $k$, and let $\mathbb{A}_k^n = Spec\, k[x_1, \ldots, x_n]$ be a piece of its standard cover. Then the point of $\mathbb{P}^n_k$ that correspond to a prime ideal $(0) \in Spec\, k[x_1, \ldots, x_n]$ has this property.