A boundary point x of a domain D that is not the endpoint of any continuous injective map γ : [0,1] → D with γ(1) = x; equivalently, there is no simple arc contained in D that approaches x, so x cannot be reached from the interior by a path without leaving the domain.

Principle

Inaccessibility is the negation of the pathwise reachability condition: a boundary point is inaccessible when no simple interior arc can land at it, regardless of sequences approaching it.

Demonstration

In a planar domain formed by removing a countable sequence of increasingly tight inward spirals accumulating at a point, the accumulation point can fail to be the endpoint of any arc lying in the domain and so is inaccessible even though it lies in the boundary and all neighborhoods intersect the domain.

Misapplication

Assuming inaccessibility implies isolation from the domain or that inaccessible points are rare; inaccessible points can be accumulation points of domain points and still be dense in parts of the boundary, so equating inaccessibility with remoteness is mistaken.

Consequence

An inaccessible boundary point limits constructions that depend on approaching along simple curves: boundary correspondence theorems that rely on curvewise access fail at inaccessible points and special techniques (e.g., prime ends with non-singleton impressions) are required.

Reversal

The reversal is an accessible point, which admits at least one simple arc from the interior landing at the point; inaccessible and accessible points represent complementary pathwise behaviors on the boundary.

Boundary

The notion requires a setting where arcs make sense (metric or pathwise topology) and applies to points of the closure that lie on the boundary; it is not a statement about isolated topological separation or measure-theoretic properties alone.

Semantic Tension

Inaccessibility contrasts with sequential approachability: a point might be a limit of interior points (sequentially approachable) yet be inaccessible by any simple curve, producing tension between pathwise and limiting notions.

Synthesis

An inaccessible point is a boundary point of a domain that cannot be reached by any simple continuous path from the interior; it highlights a topological obstruction to curvewise access even when the point is a limit of interior points.