----- > [!proposition] Proposition. ([[characterization of closure in subspace]]) > Let $Y \subset X$ be a [[subspace topology|subspace]] of [[topological space]] $X$; let $A \subset Y$. Denote by $\overline{A}$ the [[closure]] of $A$ *in $X$*. > \ > Then, the [[closure]] of $A$ in $Y$ equals $\overline{A} \cap Y$. > \ > In general, the notation $\overline{A}$ will always stand for the [[closure]] of $A$ *in $X$*, for this reason. > [!proof]- Proof. ([[characterization of closure in subspace]]) Let $B$ denote the [[closure]] of $A$ in $Y$. $\overline{A}$ is [[closed set|closed in]] $X$, so by [[characterization of closed sets in subspaces|this result]] $\overline{A} \cap Y$ is [[closed set|closed in]] $Y$. Since $\overline{A} \cap Y$ contains $A$, and $B$ is the intersection of *all* [[closed set]]s in $Y$ containing $A$ ('smallest containing closed set'), we have $B \subset \overline{A} \cap Y$. \ On the other hand, we know that $B$ is [[closed set|closed in]] $Y$. Hence by [[characterization of closed sets in subspaces]], $B=C \cap Y$ for some set $C$ [[closed set|closed in]] $X$. Since $\overline{A}$ is the intersection of *all* [[closed set]]s in $X$ containing $A$, we must have $\overline{A} \subset C$. It follows that $\overline{A} \cap Y \subset C \cap Y=B$. ----- #### ---- #### 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 > ```