Properties
- In general, dH(X,Y) may be infinite. If both X and Y are bounded, then dH(X,Y) is guaranteed to be finite.
- We have dH(X,Y) = 0 if and only if X and Y have the same closure.
- On the set of all non-empty subsets of M, dH yields an extended pseudometric.
- On the set F(M) of all non-empty compact subsets of M, dH is a metric.
- If M is complete, then so is F(M).
- (Blaschke selection theorem) If M is compact, then so is F(M).
- The topology of F(M) depends only on the topology of M, not on the metric d.
Read more about this topic: Hausdorff Distance
Famous quotes containing the word properties:
“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 (18031882)
“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 (16321704)