A nonempty subset $A$ of a topological space $X$ is called irreducible if, if $A\subset A_{1}\cup A_{2}$ and $A_{1}, A_{2}$ are closed subsets of $X$, then $A\subset A_{1}$ or $A\subset A_{2}$. We know that in a Hausdorff space $X$ a closed subset is irreducible if and only if it is a singleton.

Questions: When is a closed subset of a $T_{1}$-space irreducible? Is there an equivalent condition for it?

sober$T_1$ spaces have the property that closed irreducible subsets are singletons. Also, is your second sentence really true? Do you have a reference? $\endgroup$5more comments