Connectedness

Connectedness is an important property that a subset of a metric space may possess. Various propositions give ways of showing that a subset is connected. In \(\mathbb {R}\) , the connected subsets are precisely the intervals. Connectedness is preserved by continuous functions, a fact that is the generalization of the Intermediate Value Theorem of real analysis. Every metric space decomposes uniquely as a union of connected disjoint components. Metric spaces can vary in their connectedness properties, from being connected, with a single component, to being totally disconnected, having single points as components.