© 1996, Topology Atlas

A topological space is called rigid if the identity is its only autohomeomorphism. This can be considered a counterpart to the notion of homogeneity; a space is called homogeneous if there are many autohomeomorphisms which map any point to all over the space.

We can safely say that in a rigid space each point has a characteristic topological property (whether there is a name to it or not) in its relation to the whole space, while in a homogeneous space all points have the same property.

Topological groups are the obvious examples of homogeneous spaces. But there are no such handy type of rigid spaces. They are generally found after elaborate arguments.

Probably Sierpinski was the first to notice the existence of rigid spaces in 1932. Since then many examples have been presented for various purposes.

The famous one is the so-called Cook Curve. It was proposed in 1967 by H.Cook as a one-dimensional connected compact subspace of the three-dimensional Euclidean space, and satisfies a stronger property that every continuous map between two non-degenerate connected closed subsets is the identity (and hence, two such subsets should be identical to each other if one is mapped onto another).

Although they are found rarely, rigid spaces are neither isolated, unimportant nor pathological existence. Generally speaking, they are located almost everywhere, reasonably close (in the sense of distance and deformation) to almost every space in the universe. This is an important point in the author's view.

According to current terminology, a space is called strongly rigid if the identity and the constant maps are the only continuous maps to itself. Even such spaces are suspected to be common.

Some generalization of rigidity is sometimes considered in the literature. Let P be a particular property of continuous maps, and we call a space P-rigid if the identity and the constant maps are the only continuous maps to itself having the property P. For P, we may take closed-ness, open-ness, proper-ness and so on.

It also has to be mentioned that rigid spaces have some applications in the other branch of mathematical investigation. For example, Trnkova uses the above Cook Curve in her investigation of category theory.

Here is a question. The Cook Curve is constructed using inverse limits and multivalued maps, and is built up by adding circles at each stage. Is it possible to construct a similar space in the fashion of Cantor's middle-thirds set by removing some portions step-by-step from, say, the unit cube?