Here is an exercise from the Wedhorn's book.
Exercise 3.2. Prove that there exists a scheme which admits a covering by countably many closed subschemes each of which is isomorphic to the affine line $\mathbb{A}_{k}^{1}$ (over an algebraically closed field), indexed by $\mathbb{Z}$, such that the copies of $\mathbb{A}_{k}^{1}$ corresponding to $i$ and $i+1$ intersect in a single point, which is the point 0 when considered as a point in the $i$ -th copy, and the point 1 when considered as an element of the $(i+1)$-th copy. Prove that this scheme is connected and locally noetherian, but not quasi-compact.
I have difficulties in the first step: how to get such a scheme? I thought we should glue countable many $\mathbb A_{k}^1$ as the problem says, but a single point is not an open subset of $\mathbb A_k^1$, hence the theorem of gluing schemes doesn't work. How to modify my thoughts or could you provide a new way to get such a scheme? Thanks!