---- > [!definition] Definition. ([[box topology]]) > Let $\{ X_{\alpha} \}_{\alpha \in J}$ be an indexed family of [[topological space|topological spaces]]. Let us take as a [[basis for a topology]] on the product space $\prod_{\alpha \in J}^{} X_{\alpha}$ the collection of all sets of the form $\prod_{\alpha}^{}U_{\alpha},$ > where $U_{\alpha}$ is open in $X_{\alpha}$, for each $\alpha \in J$. The [[topology generated by a basis|topology generated by this basis]] is called the **box topology**. > [!justification] > The described collection is indeed a basis because $\prod_{\alpha \in J}^{}X_{\alpha}$ itself belongs as a basis element, and because the intersection of basis elements is again a basis element: $\left( \prod_{\alpha} U_{\alpha} \right) \cap \left( \prod_{\alpha} V_{\alpha} \right) = \prod_{\alpha}^{}(U_{\alpha} \cap V_{\alpha}).$ ---- #### ---- #### 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 > ```