---
tag: 17VG475BRV
kind: example
name: Warsaw circle
msc: [54F15, 55P55]
instance_of: [NJQ196M1WR]
uses: [0WB6DH499T, FDVAHRJKF3]
sources:
  - url: https://en.wikipedia.org/wiki/Shape_theory_(mathematics)
---

## 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|NJQ196M1WR]].
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|0WB6DH499T]], because the arc $A$ joins the vertical segment to the graph part, while $S$ alone is not path-connected.
It is not [[locally connected|FDVAHRJKF3]] 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.

## Explanations

### L1 human

The Warsaw circle is a circle with a bad spot.
One stretch of it is replaced by the graph of $\sin(1/x)$, which wiggles faster and faster up to a vertical segment.
It looks like a circle from far away and it encloses a hole.
But no loop can go around it, because a path cannot run through the infinitely many wiggles in finite time.
It shows that homotopy groups can miss a hole that other tools can see.