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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 ...
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.
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).
A topological space is said to be limit point compact if every infinite subset of has a limit point in , and countably compact if every countable open cover has a finite subcover. In a metric space , the notions of sequential compactness, limit point compactness, countable compactness and compactness are all equivalent (if one assumes the axiom ...
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).
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).
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 ...
Definition: A set is sequentially compact if every sequence {} in has a convergent subsequence converging to an element of . Theorem: A ⊆ R n {\displaystyle A\subseteq \mathbb {R} ^{n}} is sequentially compact if and only if A {\displaystyle A} is closed and bounded.