General topology Continua and dimension theory Example

Warsaw circle

Builds on Path-connected space Locally connected space

Example

Let $S = \set{(x, \sin(1/x)) : 0 < x \le 1} \cup (\set{0} \times [-1, 1])$ be the closed topologist's sine curve in the plane. The Warsaw circle $W$ is the union of $S$ with an arc $A$ from $(0, -1)$ to $(1, \sin 1)$ . The arc meets $S$ only at these two end points. For instance, $A$ can run from $(0, -1)$ down to $(0, -2)$ , across to $(1, -2)$ , and up to $(1, \sin 1)$ along straight segments.

The space $W$ is a continuum. It is closed and bounded in $\R^2$ , hence compact. It is connected, since $S$ is connected and $A$ is connected and meets $S$ .

Remarks

The space $W$ is path-connected, because the arc $A$ joins the vertical segment to the graph part, while $S$ alone is not path-connected. It is not locally connected at the points of the vertical segment. All homotopy groups of $W$ are trivial, yet $W$ is not contractible. So the map from $W$ to a point is a weak homotopy equivalence but not a homotopy equivalence. In shape theory, $W$ has the shape of the circle, and its first Čech cohomology group is $\Z$ , like that of a circle.

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Tag	`17VG475BRV`
URI	`m2://17VG475BRV`
Kind	example
Writing	W0 stub
MSC	54F15, 55P55
Instance of	Continuum
Uses	Path-connected space, Locally connected space
Changed	2 November 2026

Warsaw circle

Example

Remarks

Reading path

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