The *blue-sky catastrophe* is a space, which arises from the bifurcation of a certain dynamical system. This space may be described as the closure of a certain embedding of a cylinder in . This space can be described by starting with and allowing the clinder to circle back on , winding around towards it infinitely many times.

This space is topologically interesting because of the combination of which properties it does and does not have.

The Blue-Sky Catastrophe space can be formally described as follows: Let , be the exponential map. Let be an open arc in of diameter less than (the choice of is arbitrary, we just want to consider small intervals). For a natural number , let be the union of the connected components of , which are contained in . We define the blue-sky catastrophe space to have underlying set but with a topology coarser than the usual product topology. A basic neighborhood at a point for is a “standard one” of the form , . More interestingly, a basic neighborhood of a point is a set of the form

for an open arc of of diameter , , and natural number . Such a set looks like a square with only one boundary edge and a sequence of open cylinders of shrinking diameter approaching that boundary edge. The use of the exponential map is ensures the basic neighborhoods actually describe a coherent “winding” toward the limiting circle .

**Alternative names:** Hyperbolic smile

**Topological Properties:** 2-dimensional compact metric space. Embeds in . If you remove the limiting circle, the result is homeomorphic to an open annulus. This space is not locally path-connected at any point in the “limiting circle” .

**Homotopy and homology groups: **If has the usual product topology, then the identity function is continuous and, moreover, is a weak homotopy equivalence. In particular, is generated by a loop going around the limiting circle. Any compact subset of must be contained in for some so no loop can go along the entire length of the winding surface. All higher homotopy and homology groups are trivial.

**Homotopy Type and Shape Type:** Blue sky catastrophe is shape equivalent to the unit circle. When one “thickens” (embedding as above in ) in the construction above by any , the resulting space is homotopy equivalent to a circle. It follows that is not contractible since it is not shape equivalent to a point.

**Cech Homotopy groups:** .

Interetsingly, the groups and describe completely different things even though they are both isomorphic to . This coincidence is purely a result of using instead of some other manifold. Indeed, is generated by an actual loop going around the limiting circle while is generated not by an actual loop but by a loop going around the entire structure once is thickened by some small .

**Cech Homology groups:**

**Wild Set:** The space is semilocally simply connected **in the based sense** (for every , there exists a neighborhood of such that the homomorphism induced by inclusion is trivial). However, at the points of the limiting circle, is not semilocally simply connected **in the unbased sense** (for every , there exists a neighborhood of such that, **for every ,** the homomorphism induced by inclusion is trivial). So it depends what kind of wild set you’re defining. With the usual definition, has no wild set. But if you define to be the subspace of points at which is not semilocally simply connected in the unbased sense, then .

**Covering spaces:** The only covering map where is path-connected is the identity map. Hence, has no non-trivial *path-connected* covering spaces. However, does have non-trivial *connected* covering spaces. In particular, there is a unique covering space corresponding to each subgroup of in a way that is reminscent of the covering spaces of . In particular the trivial subgroup corresponds to the universal covering over that looks like this:

**Summary and** **Other Properties:**

**Semi-locally simply connected (in the based sense):**Yes**Semi-locally simply connected (in the based sense):**Yes**Traditional Universal Covering Space:**No, there is a universal connected covering space but not a simply connected path-connected covering space**Generalized Universal Covering Space:**Yes**Homotopically Hausdorff:**Yes**Strongly (freely) Homotopically Hausdorff:**No**Homotopically Path-Hausdorff:**Yes**:**Yes**-shape injective:**No