= Warsaw circle
{c}
The <Warsaw circle> joins the two <path components> of a closed <topologist's sine curve> by an <arc> attached at their outer endpoints. It is <path-connected> and has trivial <fundamental group>, yet admits a <continuous map> to the <circle> with no <continuous lift> to its real-line <covering space>. Set the map equal to $1$ on the closed <topologist's sine curve> and wind once on the joining <arc>. A lift would be constant on each <path component> of the sine curve, with the same constant on both by <continuity>, but would change by $1$ along the joining <arc>. This proves that the map is not <null-homotopic>, by the <homotopy lifting theorem for a covering map>. Thus finiteness of the <fundamental group> alone cannot replace <local path connectedness> in the usual <lifting criterion for a covering space>.
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