# Notes - Metric Spaces MT23, Subspaces

### Flashcards

What does it mean for one metric space $(Y, d _ Y)$ to be a subspace of another metric space $(X, d _ X)$?

$Y \subseteq X$ and $d _ Y$ is the restriction of $d _ X$ to $Y$.

Can you give an example of a set open in a subspace, but closed in the entire space?

Let $Y \subseteq X$ where $Y = \mathbb R \times \{0\}$ and $X = R^2$. Then $B _ Y(0, 1)$ is an open subset of $Y$, but a closed subset of $X$.