boundconnected

Boundconnected is a term that appears in informal mathematical discussions to describe a property related to boundedness and connectedness. It does not have a single universally accepted definition, but two common usages are observed.

One usage defines boundconnected as a property of a subset of a metric space: the subset is both bounded and connected. In Euclidean spaces, this means there exists a finite radius R such that the set lies inside a ball of radius R and the set cannot be partitioned into two nonempty separated open sets. A typical example is any closed interval [a, b] in the real line or any connected region contained within a ball of finite radius. The real line itself is not boundconnected as a whole because it is unbounded, even though its finite subintervals are.

A second usage treats boundconnected as a property of a space in which every connected component has bounded diameter. In this sense, the emphasis is on the diametric size of connected components rather than the boundedness of the entire space.

Relation to other concepts: in Euclidean space, a set that is closed and bounded is compact (by the Heine-Borel theorem), and if it is also connected, it is a compact and connected set, which is a stronger and well-defined notion. The term boundconnected, being informal, is often a shorthand for “bounded and connected,” or for the idea that connected components live within a bounded region.

Notes: because boundconnected is not a formal standard term, it is important to specify the exact properties when using it in rigorous writing. See also connected set, bounded set, compactness, and diameter.