Connected components of qcqs schemes and projective spaces

In this article, we first prove a general result in topology which states that every quasi-component of a quasi-spectral space is connected. \ As an application, the structure of the connected components of every quasi-compact quasi-separated (qcqs) scheme $X$ is fully characterized. They are exactly of the form $f^{-1}(C)$ where $f:X\to \text{Spec}(R)$ is the canonical morphism, $C$ is a connected component of $\text{Spec}(R)$ and $R={\mathcal{O}}_X(X)$ is the ring of global sections of $X$. \ Next, we make new advances in understanding the structure of the connected components of projective spaces. In general, for an $\mathbb{N}$-graded ring $R=\underset{n⩾0}{⨁}{R}_n$, the structure of the connected components of scheme $\text{Proj}(R)$ is still unknown. However, we show that for any scheme $S$ the connected components of the projective space ${\mathbb{P}}_S^n={\mathbb{P}}_{\mathbb{Z}}^n{\times }_{\text{Spec}(\mathbb{Z})}S$ are exactly of the form ${\mathbb{P}}_C^n$ where $C$ is a connected component of $S$ which is equipped with a closed subscheme structure.