Definition:Irreducible Subset

Definition
Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$ be non-empty.

Let $\tau_A$ be the subspace topology on $A$ induced by $\tau$.

Then $A$ is an irreducible subset (of $S$ in $T$) $\struct {A, \tau_A}$ is an irreducible space.