If $X$ is a topological space, let $t(X)$ be the set of nonempty irreducible closed subsets of $X$. We put a topology on $t(X)$ by saying that the closed sets are $t(Y)$, where $Y$ is closed in $X$. We have a continuous function $\alpha: X \rightarrow t(X)$ given by $\alpha(P) = \overline{\{P\}}$. It is clear that $\alpha$ is continuous, since if $E \subseteq t(X)$ is the collection of nonempty irreducible closed subsets of some closed set $Y \subseteq E$, then $$\alpha^{-1}E = \{ P \in X : \overline{\{P\}} \in E \} = \{ P \in X : \overline{\{P\}} \subseteq Y\} = Y$$ where the last equality follows because $Y$ is closed. Then Hartshorne says Note that $\alpha$ induces a bijection between open subsets of $X$ and open subsets of $t(X)$.

Why is this the case? It's just as good to give a bijection between closed sets. Is Hartshorne saying that $E \mapsto \alpha^{-1}E$ gives such a bijection? In that case, I am wondering why it is true that $Y_1 \subsetneq Y_2$ for closed sets $Y_1, Y_2$ implies that there is an irreducible closed subset of $Y_2$ which is not contained in $Y_1$.

  • $\begingroup$ What does it mean for a closed set to be irreducible closed? $\endgroup$ – DanielWainfleet Dec 26 '15 at 23:12
  • $\begingroup$ It means it's closed and irreducible? $\endgroup$ – D_S Dec 27 '15 at 1:33
  • $\begingroup$ What does irreducible mean here? $\endgroup$ – DanielWainfleet Dec 27 '15 at 1:50
  • $\begingroup$ Oh it means that the space cannot be written as the union of two proper closed sets. $\endgroup$ – D_S Dec 27 '15 at 2:20

Wait I'm dumb. If $y \in Y_2 \setminus Y_1$, then $\overline{\{y\}} \subseteq Y_2$ is such an irreducible closed set.

  • $\begingroup$ more details for silly people like me: a subspace of a topological space is irreducible if and only if its closure is irreducible. And a singleton set is of course irreducible. $\endgroup$ – D_S Dec 27 '15 at 2:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.