metric space in a sentence

Examples

1. Every metric space is Hausdorff and paracompact ( and hence normal and Tychonoff ).
2. Clearly, every isometry between metric spaces is a topological embedding.
3. Also note that any metric space is a uniform space.
4. Ostrand generalized the Kolmogorov superposition theorem to compact metric spaces.
5. An uncountable product of metric spaces need not be metrizable.
6. Let ( M, d ) be a separable metric space.
7. Not all metric spaces may be embedded in Euclidean space.
8. Also, note that any metric space is a uniform space.
9. Asymptotic cones are particular examples of ultralimits of metric spaces.
10. My question is, why is this restricted to metric spaces?