The Topologist's Sine Curve

Topology · Axiom Academy

EXAMPLE The Topologist's Sine Curve A Classic Example of Connected but Not Path-Connected The topologist's sine curve is one of the most elegant counterexamples in topology. It demonstrates that a space can be connected (impossible to separate into disjoint open sets) while simultaneously failing to be path-connected (no continuous path exists between certain points). The blue curve oscillates infinitely as it approaches the y-axis. The red segment shows its closure. The topologist's sine curve is defined as the union of two sets: is the vertical line segment from to The set is equipped with the subspace topology inherited from with the standard (Euclidean) topology. The key feature is that as approaches 0 from the right, the function oscillates infinitely between -1 and 1, getting arbitrarily close to every point on the vertical segment . Theorem 1: The Topologist's Sine Curve is Connected Recall the definition: A topological space is connected if it cannot be written as the union of two disjoint non-empty open sets. Equivalently, is connected if the only subsets that are both open and closed (clopen) are and itself. Show that is connected: The set is the graph of the continuous function over the interval . Since is connected (as an interval in ) and the continuous image of a connected space is connected, we conclude that is connected. Show that is connected: The line segment is homeomorphic to the interval , which is connected. Therefore, is connected.

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