Hausdorff Distance - Properties

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.

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