Definitions
Let be a metric space. We define a new metric on, known as the induced intrinsic metric, as follows: is the infimum of the lengths of all paths from to .
Here, a path from to is a continuous map
with and . The length of such a path is defined as explained for rectifiable curves. We set if there is no path of finite length from to . If
for all points and in, we say that is a length space or a path metric space and the metric is intrinsic.
We say that the metric has approximate midpoints if for any and any pair of points and in there exists in such that and are both smaller than
- .
Read more about this topic: Intrinsic Metric
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