Definition 111.6.9. A topological space $X$ is called irreducible if $X$ is not empty and if $X = Z_1\cup Z_2$ with $Z_1, Z_2\subset X$ closed, then either $Z_1 = X$ or $Z_2 = X$. A subset $T\subset X$ of a topological space is called irreducible if it is an irreducible topological space with the topology induced from $X$. This definition implies $T$ is irreducible if and only if the closure $\bar T$ of $T$ in $X$ is irreducible.
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