Properties
- In general, we have d ≤ dl and the topology defined by dl is therefore always finer than or equal to the one defined by d.
- The space (M, dl) is always a path metric space (with the caveat, as mentioned above, that dl can be infinite).
- The metric of a length space has approximate midpoints. Conversely, every complete metric space with approximate midpoints is a length space.
- The Hopf–Rinow theorem states that if a length space is complete and locally compact then any two points in can be connected by a minimizing geodesic and all bounded closed sets in are compact.
Read more about this topic: Intrinsic Metric
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)