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  "body": "\n## Example\n\nLet $S = \\set{(x, \\sin(1/x)) : 0 \u003c x \\le 1} \\cup (\\set{0} \\times [-1, 1])$ be the closed topologist's sine curve in the plane.\nThe *Warsaw circle* $W$ is the union of $S$ with an arc $A$ from $(0, -1)$ to $(1, \\sin 1)$.\nThe arc meets $S$ only at these two end points.\nFor instance, $A$ can run from $(0, -1)$ down to $(0, -2)$, across to $(1, -2)$, and up to $(1, \\sin 1)$ along straight segments.\n\nThe space $W$ is a [[continuum|NJQ196M1WR]].\nIt is closed and bounded in $\\R^2$, hence compact.\nIt is connected, since $S$ is connected and $A$ is connected and meets $S$.\n\n## Remarks\n\nThe space $W$ is [[path-connected|0WB6DH499T]], because the arc $A$ joins the vertical segment to the graph part, while $S$ alone is not path-connected.\nIt is not [[locally connected|FDVAHRJKF3]] at the points of the vertical segment.\nAll homotopy groups of $W$ are trivial, yet $W$ is not contractible.\nSo the map from $W$ to a point is a weak homotopy equivalence but not a homotopy equivalence.\nIn shape theory, $W$ has the shape of the circle, and its first Čech cohomology group is $\\Z$, like that of a circle.\n\n## Explanations\n\n### L1 human\n\nThe Warsaw circle is a circle with a bad spot.\nOne stretch of it is replaced by the graph of $\\sin(1/x)$, which wiggles faster and faster up to a vertical segment.\nIt looks like a circle from far away and it encloses a hole.\nBut no loop can go around it, because a path cannot run through the infinitely many wiggles in finite time.\nIt shows that homotopy groups can miss a hole that other tools can see.\n",
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