---- > [!definition] Definition. ([[irreducible topological space]]) > A [[topological space]] $X$ is said to be **irreducible** if whenever $X=X_{1} \cup X_{2}$ for [[closed set|closed subsets]] $X_{1},X_{2} \subset X$, then in fact either $X=X_{1}$ or $X=X_{2}$. > An **irreducible component** of $X$ is a maximal irreducible [[subspace topology|subspace]]. (Though maybe we are not always explicit about this in our course?) > [!equivalence] > $X$ is irreducible iff the intersection of any two nonempty open subsets $U_{1} \neq \emptyset,U_{2} \neq \emptyset$ of $X$ is nonempty: $U_{1} \cap U_{2} \neq \emptyset$. ^equivalence > [!basicproperties] > - If $X$ is irreducible, then any nonempty open subset $U \subset X$ is [[irreducible topological space|irreducible]] too, and is [[dense]] in $X$: $\overline{U}=X$. > >[!proof]- Proof. > >If we have two nonempty subsets open in $U$, $U_{1} \subset U$ and $U_{2} \subset U$, then since $U_{1}$ and $U_{2}$ are open in $X$ we have $U_{1} \cap U_{2} \neq \emptyset$. Hence $U$ irreducible. > > > >Next, Note that $X=\overline{U} \cup (X - U)$. This is a union of two [[closed set|closed subsets]], hence $X=\overline{U}$ or $X=X- U$. The latter is absurd since $U$ is nonempty. Hence $X=\overline{U}$. > >- [[closure|Closure]] preserves irreducibility: if $S \subset X$ is an [[irreducible topological space|irreducible]] [[subspace topology|subspace]] then so is $\overline{S}$. In paricular, $\overline{\{ x \}}$ is [[irreducible topological space|irreducible]] for all $x \in X$. > > [!proof]- > > Suppose $\overline{S}=C_{1} \cup C_{2}$ for $C_{1},C_{2}$ closed in $\overline{S}$. Intersect both sides with $S$: $S=S \cap \overline{S}= (S \cap C_{1}) \cup (S \cap C_{2}).$ > > $S \cap C_{1}$ and $S \cap C_{2}$ are each closed in $S$, and $S$ is irreducible, so $S=S \cap C_{1}$ or $S=S\cap C_{2}$, say, $S=S \cap C_{1}$. This means $\overline{S}=\overline{S} \cap \overline{C_{1}}=\overline{S} \cap C_{1}=C_{1}$. > - Irreducible $\implies$ [[connected]], as is clear e.g. from the equivalence. > [!basicexample] > For $A$ a [[commutative ring|commutative]] [[ring]], the irreducible closed subspaces of $\text{Spec }A$ are precisely the closed sets of the form $V(\mathfrak{p})$ for $\mathfrak{p}$ a [[prime ideal]] of $A$. See [[irreducible closed subspaces of Spec are precisely the vanishing of primes]]. ^basic-example > [!note] Remark. > The characterization above shows that irreducible topological spaces are a bit strange: for example, they are always non-[[Hausdorff space|Hausdorff]]. This may be surprising, given how the notion at first glance seems a straightforward analogue to [[connected|connectedness]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```