# Notes - Metric Spaces MT23, Isometries

### Flashcards

Suppose $X$ and $Y$ are metric spaces. What does it mean for $f : X \to Y$ to be an isometry?

\[d_Y(f(x), f(y)) = d_X(x, y)\]

for all $x, y \in X$.

Suppose we are asked to justify whether metric spaces like

- $(0, 1] \subset \mathbb R$ with metric $d(x, y) = \left \vert 1/x - 1/y\right \vert $
- $(0, \infty) \subset \mathbb R$ with metric $d(x, y) = \vert x^2 - y^2 \vert $

are complete. How can you deduce this immediately?

Completeness is preserved under isometries, take $f(x) = 1/x$ and $f(x) = x^2$. Then $[1, \infty)$ is complete, so the first example is complete, but $(0, \infty)$ is not, so the second is not complete.