Pathological (mathematics)

In mathematics, a mathematical object is called pathological if it breaks a rule that seems to be intuitively true, but is actually false. Normally, the term is used for objects specifically made to do this by mathematicians, and not simply any counterexample to an assumption.^[1]

The Alexander horned sphere is a pathological object in topology. It is a counterexample to the Jorden-Schoenflies theorem for three-dimensional objects.

The existence of a pathological object is a form of constructive proof by contradiction that shows a proposed rule does not apply to every case. The rules that are broken by pathological examples are usually called naive. Examples that obey naive rules are called well-behaved.

The term "pathological" is not formally defined, because it depends on the specific naive assumptions. Objects are only pathological in the context of a particular theory, and many pathological objects lead to the development of theories where they are well-behaved.^[2] Often, the "pathological" case is a generic property of the larger class of objects, with well-behaved objects being scarce: in this sense, being well-behaved is itself pathological.

Examples

• Many paradoxes of naive set theory involve constructing pathological sets, as in Russell's paradox and Cantor's paradox.
• The Weierstrass function is everywhere continuous but nowhere differentiable.
• The Cantor function is uniformly continuous but not absolutely continuous.
• Gabriel's horn is a solid with finite volume but infinite surface area.