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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?

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    $\begingroup$ Irreducible requires nonempty. $\endgroup$ – Martin Brandenburg Nov 29 '12 at 10:28
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    $\begingroup$ I don't think that there is an equivalent condition. After all, every affine variety (in the classical sense, i.e. without generic points) is $T_1$ and there no equivalent purely topological condition is known. $\endgroup$ – Martin Brandenburg Nov 29 '12 at 10:32
  • $\begingroup$ MArtin: topology of schemas is generally $T_0$ (sobre space). ANyway a topological space is $T_0$ iff for $x \neq y$ then $x\not\in cl(${y} or $y\not\in cl(${x}. A schema is $T_1$ iff is $T_2$ . In a $T_1$ space (i.e. such tath any singleton is closed) the only irreducible close set are the singleton. $\endgroup$ – Buschi Sergio Nov 29 '12 at 13:27
  • $\begingroup$ @Buschi Sergio: perhaps you meant to say in your last sentence that 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? $\endgroup$ – Todd Trimble Nov 29 '12 at 14:30
  • $\begingroup$ I meant third sentence about $T_2$. $\endgroup$ – Todd Trimble Nov 29 '12 at 14:30

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