Lakes of Wada

This article describes a standard counterexample to some plausible but false implications. In other words, it lists a pathology that may be useful to keep in mind to avoid pitfalls in proofs
View other standard counterexamples in topology

Lakes of Wada is used to refer to a certain compact connected subset $K$ of ${\mathbb{R}}^2$ whose complement has three connected components (all being open subsets). The three open sets here are the lakes. This example is closely related to the Jordan curve theorem.

References

• A First Course in Algebraic Topology by C Kosniowski, Cambridge University Press, Cambridge, 1980