are ‘simple’ (do not cross themselves) and ‘closed’ (have no beginning or end).
Jordan’s celebrated theorem has meaning. It states that a simple closed curve has
an inside and an outside. Its apparent ‘obviousness’ is a deception.
In Italy, Giuseppe Peano caused a sensation when, in 1890, he showed that,
according to Jordan’s definition, a filled in square is a curve. He could organize
the points on a square so that they could all be ‘traced out’ and at the same time
conform to Jordan’s definition. This was called a space-filling curve and blew a
hole in Jordan’s definition – clearly a square is not a curve in the conventional
sense.
Examples of space-filling curves and other pathological examples caused
mathematicians to go back to the drawing board once more and think about the
foundations of curve theory. The whole question of developing a better definition
of a curve was raised. At the start of the 20th century this task took mathematics
into the new field of topology.

the condensed idea

Going round the bend