A boundary point x of a domain D (typically an open connected subset of a metric or topological space) for which there exists a continuous injective map γ : [0,1] → D with γ(0) in D and γ(1) = x, i.e. x is the endpoint in the closure of D of a simple arc lying entirely in D.

Principle

Accessibility is the property of being realizable as the limit of a simple path from the interior; it is characterized by existence of a continuous arc contained in the domain that terminates at the point.

Demonstration

In the unit disk in the plane, any radial segment t ↦ (1−t)z for fixed boundary point z ∈ S^1 is a continuous arc in the disk landing at z, so every boundary point of the disk is accessible via a radial arc. In contrast, a slit plane where a boundary point lies at the tip of an inward spiral may be inaccessible.

Misapplication

Treating every limit point or accumulation point of D as accessible; a cluster point of D need not admit a simple arc from D that ends there, so equating topological closure with accessibility is incorrect.

Consequence

If a boundary point is accessible then boundary values of functions defined on D can often be interpreted by following the arc; in complex analysis this supports constructing non-tangential or curve-based boundary limits and identifying corresponding prime ends.

Reversal

The conceptual opposite is an inaccessible point, a boundary point that cannot be the endpoint of any continuous arc contained in the domain; accessibility and inaccessibility partition possible approaches by simple curves.

Boundary

Definition presumes a metric or at least path-connected open set and a recognizable closure; in purely general topological spaces without a notion of arcs or in disconnected sets the term may be meaningless or require modification.

Semantic Tension

Accessibility competes with notions of approachability by sequences or by non-simple curves: a point may be approachable by sequences inside D but fail to be accessible by any simple arc, creating tension between sequential and pathwise notions.

Synthesis

An accessible point is a boundary point of a domain that can be reached by a simple continuous curve from inside the domain; it is a topological qualification stronger than mere membership of the closure and weaker than any smoothness of the boundary.