In the introduction of the book "A History of Algebraic and Differential Topology 1900-1960" Dieudonné writes:
Ever since the concept of homeomorphism was clearly defined, the "ultimate" problem in topology has been to classify topological spaces "up to homeomorphism". That this was a hopeless undertaking was very soon apparent, the subspaces of the plane $\mathbb{R}^2$ being an obvious example.
Can you help me and clarify this "obvious example"?