Closed points of irreducible component

Question

Let $T$ be a compact (not necessarily Hausdorff) topological space and let $T=\bigcup_{i\in I} Z_i$ be a decomposition of $T$ into irreducible components. Fix some $i \in I$ and consider $Z_i$.
Is it possible that each closed point of $Z_i$ is contained in some $Z_j$, $j\neq i$?

Answer 1

It is not possible if $X$ is a Jacobson space (see the answer of Joyal in the reference which is the case if $X$ is a $k$-scheme locally of finite type, where $k$ is a field) and if the space $X$ is the union of a finite numbers of irreducible components like a Noetherian scheme.

Write $X=\bigcup_{i=1}^{i=n}X_i$ where $X_i$ is an irreducible connected component. If your hypothesis was possible, then $X'_n\subset \bigcup_{i=1}^{i=n-1}X_i$ where $X'_n$ is a set of closed points of $X_n$, since $X'_n$ is dense in $X_n$, we deduce that $X_n\subset \bigcup_{i=1}^{i=n-1}X_i$ and henceforth $X=\bigcup_{i=1}^{i=n-1}X_i$, thus $X$ would have had $n-1$ irreducible components instead of $n$.

Is the set of closed points of a $k$-scheme of finite type dense?