# 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

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. 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 $\{0\}\times [-1,1]$.

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