----- > [!proposition] Proposition. ([[connected subspace of separated set lies in one constituent]]) > Suppose $X$ is a [[topological space]] and $C \sqcup D$ is a [[separation of a topological space|separation]] of $X$. Then if $Y \subset X$ is a [[connected]] [[subspace]] of $X$, $Y$ must lie entirely within $C$ or $D$. > [!proof]- Proof. ([[connected subspace of separated set lies in one constituent]]) > $C$ and $D$ are both open in $X$, hence $Y \cap C$ and $Y \cap D$ are both open in $Y$. Since they are disjoint and union to $Y$, if both were nonempty this would constitute a [[separation of a topological space|separation]] of $Y$. Therefore, one of them is empty. ----- #### ---- #### 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 > ```