---- > [!definition] Definition. ([[isometric embedding]]) > Let $(X,d_{X})$ and $(Y, d_{Y})$ be [[metric space|metric spaces]]. If $f:X \to Y$ preserves distances: $d_{Y}\big( f(x_{1}), f(x_{2}) \big)=d_{X}(x_{1},x_{2}) \text{ for all } x_{1},x_{2} \in X,$ > then we call $f$ an **isometric embedding** of the [[topological space|topological spaces]] $X$ and $Y$. > [!justification] > We'd hope that $f$ is actually an [[topological embedding]] of [[topological space]]s. We'll show that $U \subset X$ is open in $X$ if and only if it is the preimage of an open subset of $Y$. > > $\to.$ Suppose $U \subset X$ is open in $X$. WLOG $U$ is a ball of radius $r$ about some $x_{0} \in X$ , $U=B_{d_{X}}(x_{0}, r)$. $f(U)$ is then a ball of radius $r$ about $f(x_{0})$, $f(U)=B_{d_{Y}}(f(x_{0}), r)$. Hence it is open in $Y$ and so we have that $X$ equals the preimage of an open subset of $Y$. > > $\leftarrow.$ Conversely, take $B_{d_{Y}}(y_{0}, r)$ a basic open set in $Y$. > > $f^{-1}(\{ y \in Y : d_{Y}(y_{0}, y) < r\})$ > > = $\{ x \in X : d_{Y}(f(x_{0}), f(x)) < r\}$ > > = $\{ x \in X : d_{X}(x_{1}, x_{2}) < r \}$ > > = $B_{d_{X}}(x_{0}, x)$ done. ---- #### ---- #### 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 > ```