page_title: Shape theory (mathematics)
section_title: Warsaw Circle
hierarchical_section_title: Shape theory (mathematics) / Background / Warsaw Circle
caption_attribution_description: English: This is the compactification in the plane of the graph of sin(1/x) over the interval (0,2/π], with a judiciously chosen arc from (0,-1) to (2/π,1). Created using Mathematica®.
context_page_description: Shape theory is a branch of topology, which provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory.
context_section_description: Borsuk lived and worked in Warsaw, hence the name of one of the fundamental examples of the area, the Warsaw circle. This is a compact subset of the plane produced by "closing up" a Warsaw sine curve with an arc.
   The homotopy groups of the Warsaw circle and a point are isomorphic, however the two spaces are not homotopy equivalent. Instead, the Warsaw circle is shape-equivalent to a circle. This example shows that the CW complex requirement in the Whitehead theorem is essential.
caption: The Warsaw Circle.
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