# 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:

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 .

**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.

**Other Properties:**

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

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