metric space

Definition

A metric space $(X,d)$ is a set $X$ together with a distance function $d : X \times X \to [0,\infty)$ called a metric (satisfying certain axioms) defined upon it.

Notes

• can always define a Borel measure in a metric space using the sigma algebra generated by open sets.
• metric spaces are Hausdorff ($T_2$)

References

1. https://ncatlab.org/nlab/show/metric+space
2. https://en.wikipedia.org/wiki/Metric_space
3. https://math.stackexchange.com/questions/1402847/whats-the-relationship-between-a-measure-space-and-a-metric-space
4. https://unapologetic.wordpress.com/2010/03/24/measure-as-metric/
5. CSC2414 - Metric Embeddings∗ Lecture 1: A brief introduction to metric embeddings, examples and motivation https://www.cs.toronto.edu/~avner/teaching/S6-2414/LN1.pdf
6. https://proofwiki.org/wiki/Metric_Space_is_Hausdorff