A point x in a topological space X for which no neighborhood of x is homeomorphic to any Euclidean space R^n (or half-space R^n_+) of a fixed dimension; equivalently, x is a local failure of manifold structure and cannot be given a single consistent local chart model.

Principle

A manifold requires uniform local Euclidean (or half-Euclidean for boundary points) neighborhoods; a nonmanifold point violates that uniformity at the local scale and so blocks the existence of an atlas covering X that makes X a manifold.

Demonstration

The cone apex in a 2-dimensional cone (the tip) is a nonmanifold point for the cone regarded as a subspace of R^3: every neighborhood of the apex is homeomorphic to a cone neighborhood, not to an open disk in R^2, so no 2-dimensional chart exists there.

Misapplication

Calling every isolated singularity or branch point a manifold point because it lies in an otherwise manifold set; mistakenly treating a vertex of a graph embedded in the plane as a manifold point and applying manifold theorems that require local Euclidean charts.

Consequence

Topological and differential tools that rely on local Euclidean structure fail at the point: there is no local coordinate chart, standard invariants like tangent spaces are undefined, and manifold-level classification and extension theorems may not apply.

Reversal

A manifold point: a point whose neighborhood is homeomorphic to R^n (or R^n_+ for a boundary point), supporting local charts and the usual manifold apparatus.

Boundary

Applies to points in general topological spaces or subsets of Euclidean spaces; excludes points that admit any single Euclidean or half-Euclidean neighborhood. Does not by itself determine global pathologies or singularity types beyond local chart failure.

Semantic Tension

The tension is between 'singularity' viewed as a geometric or analytic irregularity and 'nonmanifold' as a purely topological absence of local Euclidean charts; some analytic singularities may still be manifold points topologically, and some topological nonmanifold points may have mild analytic descriptions.

Synthesis

A nonmanifold point is a local obstruction to manifold structure: it is a point whose neighborhoods cannot be modeled on Euclidean space of any fixed dimension, so local chart-based manifold theory cannot be applied there.