---- > [!definition] Definition. ([[sober topological space]]) > A [[topological space]] $X$ is said to be **sober** if every [[irreducible topological space|irreducible]] [[closed set|closed subset]] $Z$ has a unique **generic (i.e., [[dense]]) point** $\eta \in Z$, that is, a unique $\eta \in Z$ [[closure|such that]] $\overline{\{ \eta \}}=Z$. ^definition > [!basicproperties] > - If $X$ is sober, then so is any open subset $U \subset X$. ^properties ---- #### ---- #### 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 > ```