Warsaw Circle

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:

W=\{(x,\sin(\pi/x)\mid 0<x<1\}\cup \left(\{0\}\times [-1,1]\right)\cup \{(x,y)\mid (x - 1/2)^2 + (y + 15/16)^2 = (17/16)^2,y\leq 0\}

The Warsaw circle. The blue vertical arc is a limiting set of points for the topologists sine curve. The circular arc is attached to make the space path connected.

Other names: Polish circle

Topological Properties:  Planar, 1-dimensional, path-connected, compact metric space. This space is not locally path-connected at any point in the arc \{0\}\times [-1,1].

Homotopy and homology groups: The space W is weakly homotopy equivalent to a point, i.e. \pi_n(W) and H_n(W) is trivial for all n\geq 0.

Homotopy Type and Shape Type: The Warsaw circle is shape equivalent to the unit circle. When one “thickens” W in the construction above by any \epsilon >0, the resulting space is homotopy equivalent to a circle. It follows that W 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: \check{\pi}_n(W)\cong\begin{cases}  \mathbb{Z}, & n=1 \\ 0, & n \neq 1   \end{cases}

Cech Homology groups: \check{H}_n(W)\cong\begin{cases}  \mathbb{Z}, & n=1 \\ 0, & n \neq 1   \end{cases}

Wild Set: The space W is locally simply connected in the weak sense that every w\in W has a neighborhood base of open sets U for which every path-component of U is simply connected.

Other Properties:

  • Semi-locally simply connected: Yes
  • Traditional Universal Covering Space: Yes, it is simply connected so it is its own universal covering space