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In fact, every compact metric space is a continuous image of the Cantor set. Consider the set K of all functions f : R {\displaystyle \mathbb {R} } → [0, 1] from the real number line to the closed unit interval, and define a topology on K so that a sequence { f n } {\displaystyle \{f_{n}\}} in K converges towards f ∈ K if and only if { f n ...
A metric space (,) is said to have the Heine–Borel property if each closed bounded [7] set in is compact. Many metric spaces fail to have the Heine–Borel property, such as the metric space of rational numbers (or indeed any incomplete metric space).
Hausdorff and Gromov–Hausdorff distance define metrics on the set of compact subsets of a metric space and the set of compact metric spaces, respectively. Suppose (M, d) is a metric space, and let S be a subset of M. The distance from S to a point x of M is, informally, the distance from x to the closest point of S.
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, [1] [2] measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie Pompeiu ...
A metric space is said to be totally bounded if every sequence admits a Cauchy subsequence; in complete metric spaces, a set is compact if and only if it is closed and totally bounded. [2] Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded).
Some examples of spaces that are not limit point compact: (1) The set of all real numbers with its usual topology, since the integers are an infinite set but do not have a limit point in ; (2) an infinite set with the discrete topology; (3) the countable complement topology on an uncountable set. Every countably compact space (and hence every ...
Most commonly X is called locally compact if every point x of X has a compact neighbourhood, i.e., there exists an open set U and a compact set K, such that . There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular).
Every compact metric space is complete, though complete spaces need not be compact. In fact, a metric space is compact if and only if it is complete and totally bounded. This is a generalization of the Heine–Borel theorem, which states that any closed and bounded subspace of R n is compact and therefore complete. [1] Let (,) be a complete ...