Connectedness from global regular functions
= Connectedness from global regular functions
If a nonempty <scheme> $X$ has no nontrivial <idempotent> in $\Gamma(X,\mathcal O_X)$, then its underlying <topological space> is <connected>. Indeed, a decomposition into two nonempty clopen subsets would produce the global regular function equal to zero on one component and one on the other. In particular, $\Gamma(X,\mathcal O_X)$ being a <field> implies connectedness.