----- > [!proposition] Proposition. ([[closure and union]]) > Let $A$, $B$, and $A_{\alpha}$ denote subsets of a [[topological space]] $X$. > **If $A \subset B$, then $\overline{A} \subset \overline{B}$.** \ Suppose $A \subset B$ and let $\overline{a} \in \overline{A}$. Then $\overline{a}$ is in the intersection of all [[closed set]]s containing $A$. $\overline{B}$ is a [[closed set]] containing $A$; thus, it contains that intersection. Hence $\overline{a} \in \overline{B}$. \ **$\overline{A \cup B}=\overline{A} \cup \overline{B}$.** (Extends to finite union.) \ Let $x \in \overline{A \cup B}$. Then for every [[neighborhood]] $U \ni x$, we have $U \cap (A \cup B) \neq \emptyset$. This implies that $U \cap A$ or $U \cap B$ is nontrivial, meaning that one if $x \in \overline{A}$ or $x \in \overline{B}$ is true. Hence $\overline{A \cup B} \subset \overline{A} \cup \overline{B}$. Conversely let $x \in \overline{A} \cup \overline{B}$. Then for every [[neighborhood]] $U \ni x$ we have $U \cap A \neq \emptyset$ or $U \cap B \neq \emptyset$. Then $U \cap (A \cup B) \neq \emptyset$ so $x \in \overline{A \cup B}$. \ **$\overline{\bigcup_{}^{}A_{\alpha} }\supset \bigcup_{}^{}\overline{A}_{\alpha}$.** \ Let $x \in {\bigcup_{}\overline{A}_{\alpha}}$. Then for some $A_{\alpha}$ we have that for every [[neighborhood]] $U \ni x$, $U \cap A_{\alpha} \neq \emptyset$. Since $A_{\alpha} \subset \bigcup_{}^{}A_{\alpha}$, we in turn have that $U \cap \bigcup_{}^{}A_{\alpha} \neq \emptyset$, implying that $x \in \overline{\bigcup_{}^{}A_{\alpha} }$. \ *Warning: The reverse inclusion may fail!* For example, let $A$ be the collection of sets for which each element is a singleton containing a member of $\mathbb{Q}$. Then $\bigcup_{\alpha}^{}A_{\alpha}=\mathbb{Q}$ and therefore $\overline{\bigcup_{\alpha}^{}A_{\alpha}}=\mathbb{R}$. But, since each member of $A$ is a [[closed set]], we have $\bigcup_{}^{}\overline{A}_{\alpha}=\mathbb{Q} \neq \mathbb{R}.$ The reverse inclusion will hold in the case of [[locally finite|local finiteness]], see [[closure characterization of local finiteness]]. ^48a094 ----- #### ---- #### 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 > ```