### Metric Spaces

##### due by Thursday, Sep 8, 2022

Let `$(X, d)$`

be a metric space. Define a function `$\bar{d}: X \times X \to \R$`

by `$\bar{d}(p, q) = \min(d(p, q), 1)$`

. Recall that the metric space `$(X, d)$`

is said to be bounded if there exits `$M\in \R$`

such that for all points `$p, q\in X$`

we have `$d(p, q) < M$`

- Show that
`$\bar{d}$`

is a metric on`$X$`

. **Prove or disprove:**The metrics`$d$`

and`$\bar{d}$`

induce the same topology on`$X$`

.**Prove or disprove:**The metrics`$d$`

and`$\bar{d}$`

are Lipschitz equivalent if and only if`$(X, d)$`

is bounded.