----- > [!proposition] Proposition. ([[product and subspace topologies commute]]) > If $A$ is a [[subspace topology|subspace]] of $X$ and $B$ a [[subspace topology|subspace]] of $Y$, then the [[product topology]] on $A \times B$ is the same as the [[topological space|topology]] $A \times B$ inherits from $X \times Y$. > [!proof]- Proof. ([[product and subspace topologies commute]]) This is just a matter of writing out definitions. > Let $U \times V$ denote a general [[basis for a topology|basis]] element of $X \times Y$, where $U$ is [[open set|open in]] $X$ and $V$ is [[open set|open in]] $Y$. First consider the general [[basis for the subspace topology|basis element]] for the [[product topology]] $\tau_{A \times B}$ on $A \times B$: $\begin{align} > B_{A \times B} > = & \{ (A \cap U) \times (B \cap V) : U \in \tau_{X} \and V \in \tau_{Y}\}. > \end{align}$ > \ > Then consider the general basis element for the [[topological space|topology]] $A \times B$ inherits as a [[subspace topology|subspace]] of $X \times Y$: $\begin{align} > B_{{A \times B}}^{\text{inherited}} > =& \{ (A \times B) \cap (U \times V) : U \in \tau_{X} \and V \in \tau_{Y}\}. > \end{align}$ > Observing that $(A \times B) \cap (U \times V)=(A \cap U) \times (B \cap V)$ we conclude the two [[basis for a topology|basis elements]] are equal, and thus the topologies generated by them are too. ----- #### ---- #### 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 > ```