----- > [!proposition] Proposition. ([[characterization of closed sets in subspaces]]) > Let $Y$ be a [[subspace topology|subspace]] of $X$. Then a set $A$ is [[closed set|closed in]] $Y$ if and only if it equals $C \cap Y$ for some set $C$ [[closed set|closed in]] $X$. > [!proof]- Proof. ([[characterization of closed sets in subspaces]]) > Assume that $A=C \cap Y$, where $C$ is [[closed set|closed in]] $X$. We want to show $Y \cut A$ is [[open set|open in]] $Y$. By assumption $X \cut C$ is [[open set|open in]] $X$, hence $Y \cap (X \cut C)$ is [[open set|open in]] $Y$. But $Y \cap (X \cut C) = Y \cut A$ (draw a picture). \ Conversely, assume that $A$ is [[closed set|closed in]] $Y$. Then $Y \cut A=Y \cap U$ for some $U$ [[open set|open in]] $X$. $X \cut A$ is [[closed set|closed in]] $X$ but $A=Y \cap (X \cut U)$— we're done. ----- #### ---- #### 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 > ```