----- > [!proposition] Proposition. ([[continuous on product space iff continuous on coordinates]]) > Let $X$ and $X_{i}$, $i \in I$, be [[topological space|topological spaces]]. A function $f: X \to \prod_{i \in I}^{} X_{i}$ is [[continuous]] wrt the [[product topology]] on $\prod_{i \in I}^{}X_{i}$ if and only if all of its coordinate functions are [[continuous]]. > [!proof]- Proof. ([[continuous on product space iff continuous on coordinates]]) > With $J=j_{1},\dots,j_{n}$ a finite index set, let $U=\left( \prod_{k=1}^{n}U_{j_{k}} \right) \times (\prod_{i \in I\cut J}^{}X_{i})$ be an arbitrary open set in the [[product topology]]. > > $\to.$ Suppose all coordinate functions are [[continuous]]. It suffices to show that inverses images of [[basis for a topology|basic open sets]] in the product space are open in $X$. An element $x \in X$ lies in $f ^{-1}(U)$ if and only if $f_{j_{1}}(x) \in U_{j_{1}}\text{ and } \dots \text{ and } f_{j_{n}}(x) \in U_{j_{n}} \text{ and } f_{i}(x) \in X_{i} \ \fa i.$ > Of course, the last requirement is trivially, so we just have that $x \in f^{-1}(U) \text{ iff } x \in \bigcap_{k=1}^{n}f_{j_{k}}^{-1}(U_{j_{k}}),$ > thus, $f^{-1}(U)= \bigcap_{k=1}^{n}f_{j_{k}}^{-1}(U_{j_{k}}).$ > This is a finite intersection of open sets in $X$, since each $f^{-1}(U_{j_{k}})$ is open in $X$ by construction. So it is open in $X$ and this direction is complete. > > $\leftarrow.$ Suppose $f$ is [[continuous]]. Then $f^{-1}(U)=f^{-1}\left(( \prod_{k=1}^{n}U_{j_{k}} \right) \times (\prod_{i \in I\cut J}^{}X_{i}))$ is open in $X$. > If $f_{i}^{-1}(U_{i})$ not open in $X$, then $U_{i}$ not open in the product space. Same for $f_{j_{k}}^{-1}(U_{j_{k}})$ and $U_{j_{k}}$. But these *are* open in the product space. So the inverse images of coordinate functions must be open. > > > ----- #### ---- #### 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 > ```