----- > [!proposition] Proposition. ([[product of factor bases is basis of product topology]]) > If $\mathscr{B}$, $\mathcal{C}$ are a [[basis for a topology|bases for the topologies of]] sets $X,Y$, then the set $\mathscr{D}:=\{ B \times C : B \in \mathscr{B}, C \in \mathcal{C} \}$is a [[basis for a topology|basis]] for the [[product topology|product]] [[topological space|topology]] on $X \times Y$. > More generally, suppose we have a family of [[topological space|topologies]] $\{ X_{\alpha} \}_{\alpha \in J}$ for $J$ an index set. Suppose the [[topological space|topology]] on each space $X_{\alpha}$ is given by a [[basis for a topology|basis]] $\mathscr{B}_{\alpha}$. Then the collection of all sets of the form $\prod_{\alpha \in J}^{}B_{\alpha},$ where $B_{\alpha} \in \mathscr{B}_{\alpha}$ for finitely many indices and $B_{\alpha}=X_{\alpha}$ for all the remaining indices, will serve as a basis for the [[product topology]] $\prod_{\alpha \in J}^{} X_{\alpha}$. > [!proof]- Proof. ([[product of factor bases is basis of product topology]]) > [[condition for obtaining a basis from a topology|We'll show that]] $\mathscr{D}$ [[nestles in]] $\tau_{\text{product}}$. Given an [[open set]] in $U \times V$ in $\tau_{\text{product}}$, where $U \in \tau_{X}$ and $V \in \tau_{Y}$, let $(x,y) \in U \times V$. Since $U$ [[open set|open in]] $X$, there exists $B \in \mathscr{B}$ s.t. $x \in B \subset U$. Likewise, there exists $C \in \mathcal{C}$ s.t. $x \in C \subset V$. This implies $(x,y) \in B \times C \subset U \times V,$ so $\mathscr{D}$ is a [[basis for a topology|basis for the]] [[product topology]] on $X \times Y$ by [[condition for obtaining a basis from a topology]]. ----- #### ---- #### 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 > ```