I am reading Wedhorn's Algebraic Geometry. On the page 88, it says:
Example 3.45. If $k$ is a field, and $X$ is a $k$ -scheme of finite type, then all subschemes of $X$ are of finite type over $k$. Indeed, if $X$ is affine, then this is obvious for principal open subsets of $X$; this shows that the statement is true for arbitrary open subschemes of a $k$ -scheme of finite type...
Up to here, it's pretty easy to show that every open subscheme of a $k$-scheme of finite type, is itself locally of finite type, but how to show that they are quasi-compact? (In this book, a scheme of finite type should be quasi-compact.)
For an arbitrary finitely generated $k$-algebra $A$, can we prove that every open subset of $Spec A$ is quasi-compact? (It seems to be right from the example in this book) Could you give a proof? Thanks!