An isolated boundary or image point at which k distinct local branches or sheets of a mapping or of a geometric object meet for some integer k ≥ 3, producing an intersection of multiplicity k rather than a simple transverse double point.

Principle

A multiple point is classified by the number of distinct local preimage branches meeting at the image and by their mutual tangency and intersection orders; multiplicity counts geometric sheets rather than algebraic coincidence alone.

Demonstration

A plane configuration formed by three distinct smooth arcs or lines crossing at a single point (for example, three lines through the origin with pairwise distinct directions) produces a Triple Multiple Point where three local branches meet at that point.

Misapplication

Using 'Multiple Point' to label any complicated neighborhood or dense self-intersection, or conflating a transverse double point (node) with a genuine multiple point of multiplicity three or more.

Consequence

Multiple points change local topological type and intersection numbers, affect invariants such as local contribution to genus or linking, and constrain allowable deformations; their resolution often requires blowing up or local normalization.

Reversal

The opposite concept is a simple transverse double point (node) or a regular point: situations in which at most two distinct local branches meet transversely or no branching occurs at all.

Boundary

Pertains to isolated intersections of smooth or analytic branches where the number of distinct local sheets is finite and well-defined; excludes accumulation points of intersections, nonisolated singular loci, and pathological nonmanifold behaviors.

Semantic Tension

Competes with notions of 'high-multiplicity singularity' that emphasize algebraic multiplicity of a defining function; here the emphasis is on distinct geometric branches meeting, which may differ from algebraic multiplicity in nonreduced settings.

Synthesis

A Multiple Point is an isolated image point where three or more distinct local geometric branches meet; it is measured by branch count and contact orders, has definite topological and enumerative consequences, and is distinct from simple double nodes or nonisolated intersections.