----- > [!proposition] Proposition. ([[closure and intersection]]) > Let $A,B,$ and $A_{\alpha}$ denote subsets of a [[topological space]] $X$. > **$\overline{A \cap B} \subset \overline{A} \cap \overline{B}$.** > Let $c \in \overline{A \cap B}$. Then $c$ is in the intersection of all [[closed set]]s containing $A \cap B$, hence in $\overline{A}$ and in $\overline{B}$. > *The reverse inclusion is not true!* Consider the [[open set]]s $(0,1)$ and $(1,2)$ in $\mathbb{R}$ with the [[standard topology on the real line|standard topology]]. > More generally, we can say that: > **$\overline{\bigcap_{\alpha}^{} A_{\alpha} } \subset \bigcap_{\alpha}^{} \overline{A_{\alpha}}$.** > Let $c \in \overline{\bigcap_{\alpha}^{} A_{\alpha} }$. Then $c$ belongs to the intersection of all [[closed set]]s containing $\bigcap_{\alpha}^{}A_{\alpha}$; in particular it belongs to each $\overline{A_{\alpha}}$. > **There is no relationship between $\overline{A - B}$ and $\overline{A}-\overline{B}$.** To see this, consider $\mathbb{R}$ with the [[standard topology on the real line|standard topology]]. Letting $\mathbb{I}$ denote the irrational real numbers, we have that $\overline{\mathbb{R}-\mathbb{I}}=\overline{\mathbb{Q}}=\mathbb{R}$, which is not equal to $\overline{\mathbb{R}}-\overline{\mathbb{I}}= \emptyset$. So $\overline{A-B} \not \subset \overline{A} - \overline{B}$. On the other hand, letting $A=(0,1)$ and $B=(1,2)$ we see that $\overline{A}-\overline{B}=[0,1) \neq [0,1]= \overline{A-B}$. So $\overline{A}-\overline{B} \neq \overline{A-B}$. ^a21f89 ----- #### ---- #### 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 > ```