TopologikerSinusKurve

TopologikerSinusKurve, commonly called the Topologist's Sine Curve, is a standard object in topology used to illustrate subtle properties of connectedness, local connectedness, and path-connectedness. It is defined as the closure in the plane of the graph of the function sin(1/x) for x > 0. Equivalently, the set consists of all points (x, sin(1/x)) with x > 0 together with the limit points accumulated at x = 0, namely the vertical segment {0} × [-1, 1].

Properties of the construction include that it is connected but not path-connected. The portion with x > 0 forms the oscillating curve, which accumulates onto the vertical segment at x = 0. There is no continuous path within the set that connects a point with x > 0 to a point on the segment x = 0. The origin is a point of non-local connectedness. The set is closed in the plane but unbounded in the x-direction while bounded in the y-direction.

Variants and significance: Generalizations replace sin(1/x) by other oscillatory functions such as sin(1/x^p), or alter amplitudes and domains. The construction is a classical textbook example used to demonstrate that connectedness does not imply path-connectedness, and that closures can introduce unexpected local structure, challenging intuition about continuity and connectivity in the plane.

Historical notes: The Topologist's Sine Curve appears in many standard topology references as a minimal pathological example illustrating these distinctions.

See also: Topologist's sine curve, connectedness, path-connectedness, local connectedness, sine curve.