---- > [!definition] Definition. ([[standard bounded metric]]) > Let $(X,d, \tau)$ be a [[metric space]] with [[topological space|topology]] $\tau$ induced by $d$. The **standard bounded metric** on $X$ is defined $\begin{align} \overline{d}: & X \times X \to \mathbb{R} \\ \overline{d}(x,y)= & \min \{ d(x,y), 1 \}. \end{align}$ It [[metric space|induces]] $\tau$, just as $d$ does. > [!justification] > First we show $\overline{d}$ is a [[metric]]. The first two properties are immediate. The [[triangle inequality]] holds because, letting $x,y,z \in X$, we have $\begin{align} \overline{d}(x,z)= & \min \{ d(x,z), 1 \} \\ \leq & \min \{ d(x,y) + d(x,z), 1 \} \\ \leq & \min \{ d(x,y) ,1 \} + \min \{ d(y,z), 1 \} \\ = & \overline{d}(x,y) + \overline{d}(y,z). \end{align}$ (Verify each step in your head.) > Now we note that in any [[metric space]], the collection of $\varepsilon$-balls with $\varepsilon<1$ forms a [[basis for a topology|basis]] for the [[metric 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 > ```