A cut point of a topological space is a point whose removal increases the number of connected components of the space; in other words, removing that single point disconnects a previously connected space.

Principle

Detect global decomposability by single-point removal: a cut point witnesses that the space is not irreducibly connected at that location.

Demonstration

In the closed interval [0,1], any interior point x in (0,1) is a cut point because [0,1] \ {x} splits into two disjoint open intervals. In contrast, no point of a circle S^1 is a cut point because removal of any single point leaves the circle connected.

Misapplication

Calling a boundary point of a disconnected space a cut point without checking connectivity hypotheses, or conflating cut points with arbitrary points whose removal disconnects only after also removing other points; the notion requires the removal of that single point to increase component count.

Consequence

Locating cut points yields a decomposition of the space into blocks or pieces connected via articulation points; in graph-theoretic settings cut points (articulation vertices) guide algorithms for biconnected components and influence homotopy and fundamental group structure.

Reversal

A non-cut point (cut-free point) is a point whose removal leaves the number of connected components unchanged; such points do not serve as single-point articulations of the space.

Boundary

Definition is meaningful only relative to an assumed connected ambient space; in totally disconnected spaces every point may be a cut point or none may meaningfully increase components depending on the topology. The notion is distinct from multi-point separations and from separation by closed sets of higher cardinality.

Semantic Tension

Tension exists between 'cut point' in general topology and related graph-theoretic 'articulation point' or manifold-theoretic removal phenomena; subtle differences arise when local versus global connectivity, and when working with non-Hausdorff or pathological spaces.

Synthesis

A cut point is a single point whose deletion splits a connected space into more components, serving as an atomic articulation that reveals decomposability and guides both combinatorial and topological analyses of connectivity.