The Warsaw circle is a well-known path-connected space that involves the usual topologist’s sine curve. This space provides a fundamental example in shape theory. Specifically, one many construct it as:
Other names: Polish circle. The names “Warsaw” and “Polish” honor the many contributions of Polish mathematicians to topology, e.g. Karol Boruk’s development of shape theory.
Topological Properties: Planar, 1-dimensional, path-connected, compact metric space. This space is not locally path-connected at any point in the arc .
Homotopy and homology groups: The space is weakly homotopy equivalent to a point, i.e. and is trivial for all .
Homotopy Type and Shape Type: The Warsaw circle is shape equivalent to the unit circle. When one “thickens” 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. This provides a counterexample of Whitehead’s theorem in the category of compact metric spaces.
Cech Homotopy groups:
Cech Homology groups:
Wild Set: The space is locally simply connected in the weak sense that every has a neighborhood base of open sets for which every path-component of is simply connected.
- Semi-locally simply connected: Yes
- Traditional Universal Covering Space: Yes, it is simply connected so it is its own universal covering space