Some metrics satisfy a stronger version of the triangle inequality: # For all x, y, z in M, d(x, z) <= max(d(x, y), d(y, z)) These metrics are called super-metrics. An equivalent condition is that every triangle has at least two equal sides. |
* The p-adic numbers are a super-metric space. |
Some metrics satisfy a stronger version of the triangle inequality: # For all x, y, z in M, d(x, z) <= max(d(x, y), d(y, z)) These metrics are called super-metrics. An equivalent condition is that every triangle has at least two equal sides. The p-adic numbers are a super-metric space. |
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The distance function is usually called a metric. The triangle inequality means that if you go from x to z directly, that is no longer than going first from x to y, and then from y to z. In Euclidean geometry, this is easy to see. Metric spaces allow this concept to be extended to a more abstract setting.
In any metric space M we can define the open balls as the sets of the form
Metric spaces are normal Hausdorff spaces; an important consequence is that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space. It is also true that any real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space.
A metric space in which every Cauchy sequence converges is said to be complete. A metric space is bounded if it is equal to some open ball. It is totally bounded if for every r > 0 it is the union of finitely many open balls of radius r. It is not difficult to see that every totally bounded metric space is bounded. It can be shown that a metric space is compact if and only if it is complete and totally bounded.
By restricting the metric, any subset of a metric space is a metric space itself. We call such a subset complete, bounded, totally bounded or compact if it, considered as a metric space, has the corresponding property.
Some metrics satisfy a stronger version of the triangle inequality:
These metrics are called super-metrics. An equivalent condition is that every triangle has at least two equal sides. The p-adic numbers are a super-metric space.