In Pugh's analysis book, why are these metric spaces? Pugh introduces the notion of metric space in chapter 2 as follows

Definition: A metric space is a set $X$ equipped with a metric $d$

Clear!

For example, a metric space is $(\mathbb{R}, |x-y|)$, where $|\cdot|$
  is the absolute value

Super clear!

The following are also metric spaces


What!!!!!??? It is my first time doing analysis...
Can someone please illustrate just for a single example in the above figure as to what the set and the metric would be. For example, what would be the set $X$ and the metric $d$ used in the "closed spiral" figure?
Much thanks!

You can consider all of these sets as subsets of the euclidean plane, which is a metric space with euclidean distance $d$. Restricting $d$ to these each of these sets provides a metric for each of them.
