Assertion: metric space | /r/HasProperty | complete
Facets:
value: in mathematical analysis
type: OTHER
Source sentences:
text: A metric space M {\displaystyle M} is said to be complete if every Cauchy sequence converges in M {\displaystyle M} .
text: A metric space $(X,\rho)$ is called complete if each Cauchy sequence in it converges to a point of it.
text: A metric space is compact if and only if it is complete and totally bounded.
text: Completeness of a metric space is not a topological property: A metric space homeomorphic to a complete metric space may be non-complete, for example, the real line
text: 0<x<1\}$ with the same metric; however, the first metric space is complete and the second is not.
text: However, the following characteristic property holds: A metric space is compact if and only if every metric space homeomorphic to it is complete.
text: In fact, a metric space is compact if and only if it is complete and totally bounded.
text: A metric space is complete if and only if its expansion constant is ≤ 2.
text: A metric space $X$ is complete if every Cauchy sequence $a_n \in X$ converges to an element $a \in X$.
text: However the following statement is false: "If a metric space is complete and has finite diameter, then it is compact.
text: A metric space is compact if and only if it is complete and totally bounded.
text: The b-metric space is complete if every Cauchy sequence converges.
text: A metric space $X$ is complete if every Cauchy sequence in $X$ converges in $X$.
text: A metric space is complete if every Cauchy sequence converges.
text: I know a metric space is complete if every Cauchy sequence is convergent.
text: For any p ≥ 1, the metric space (P p (M), W p ) is separable, and is complete if (M, d) is separable and complete.
text: Complete A metric space is complete if every Cauchy sequence converges.
text: A metric space is compact if and only if it is complete and totally bounded.
text: In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.
text: A metric space is called complete if all Cauchy sequences converge.