Connectedness

Connectedness captures the idea that a space is all of one piece — it cannot be split into two disjoint nonempty open parts — and is the topological reason the intermediate value theorem holds.

Definition

A topological space is connected if it cannot be written as the union of two disjoint nonempty open sets; it is path-connected if any two of its points are joined by a continuous path.

Scope

This topic defines connectedness and the related, stronger notion of path-connectedness, together with their local versions (local connectedness, local path-connectedness). It covers connected components and path components, the behavior of connectedness under continuous images, products, and unions, and the standard separating examples such as the topologist's sine curve where connected and path-connected diverge. The generalization of the intermediate value theorem to connected spaces is included.

Core questions

• What is the precise difference between connectedness and path-connectedness, and when do they coincide?
• How do connected components partition an arbitrary space, and why are they closed?
• Why is the continuous image of a connected space connected, and how does this generalize the intermediate value theorem?
• How do local connectedness and local path-connectedness control the structure of components?

Key concepts

• Connected and disconnected spaces
• Path-connectedness and path components
• Connected components and quasi-components
• Local connectedness and local path-connectedness
• Intermediate value theorem as a connectedness statement

Clinical relevance

Connectedness is the foundation for counting pieces of a space and is the degree-zero shadow of homotopy and homology; path-connectedness is the prerequisite for a well-defined fundamental group, linking general topology to algebraic topology.

History

The intuitive idea of a space being in one piece was made precise in the early 20th century alongside the axiomatization of topological spaces; the careful separation of connectedness from path-connectedness, illustrated by examples like the topologist's sine curve, became a standard part of the point-set curriculum.

Key figures

• Camille Jordan
• Felix Hausdorff
• James Munkres

Seminal works

• munkres2000
• kelley1955

Frequently asked questions

Does connected imply path-connected?

No. The topologist's sine curve is connected but not path-connected. The converse does hold: every path-connected space is connected.

When are connected and path-connected components the same?

In a locally path-connected space, connected components and path components coincide and are open, which is why manifolds and open subsets of Euclidean space behave so simply.