Assertion: metric space | /r/HasProperty | compact
Facets:
value: further
type: DEGREE
value: locally
type: MANNER
Source sentences:
text: A metric space M is compact if every sequence in M has a subsequence that converges to a point in M. This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and compactness defined via open covers.
text: A metric space is compact if and only if it is complete and totally bounded.
text: A metric space is said to be locally compact if every point has a compact neighborhood.
text: Further, a metric space is compact if and only if each real-valued continuous function on it is bounded (and attains its least and greatest values).
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: However the following statement is false: "If a metric space is complete and has finite diameter, then it is compact.
text: Every metric space that is complete (i.e. every Cauchy sequence of points in the space converges to a point within the space) and totally bounded is compact.
text: A metric space is compact if and only if it is complete and totally bounded.
text: A metric space is compact if and only if it is complete and totally bounded.