Edward’s Continuum

Edward’s Continuum is an example of a contractible continuum (compact, connected metric space) that is not locally connected at any of its points. The construction is apparently attributed to Robert D. Edwards [1].

Step 1: To construct this space, let $X\subseteq [0,1]^2$ be the following arrangement of Cantor fans: Let $C$ be the ternary Cantor set and $C_n= \{\frac{x}{2^{n-1}}\mid x\in C \}$ be a Cantor set in $[0,\frac{1}{2^{n-1}}]$ . Let $F(n)$ be the union of all the line segments from $x_n=\left(1-\frac{1}{2^{n-1}},0 \right)$ to the points in $\{1-\frac{1}{2^{n}}\}\times C_n$ . Define $x_0=(1,0)$ and set $X=\bigcup_{n\geq 1}\cup \{(1,0)\}$ . We take $x_0$ to be the basepoint of $X$ .

Let $B=[0,1]\times\{0\}$ be the base-arc of $X$ . It’s straightforward to see that $X$ deformation retracts onto $B$ . From this we can see that $X$ is contractible and that it is only locally (path) connected at the left-most point $x_1=(0,0)$ .

The space X, where a copy of a shrinking sequence of Cantor fans is attached to the unit interval.

Step 2: Now let $Y$ be the reduced suspension $\Sigma X$ of $X$ , that is,

$Y=X\times [0,1]/(X\times\{0,1\}\cup \{x_0\}\times [0,1])$

Let $y_0$ be the canonical basepoint of $Y$ . Actually, the space $Y$ can be constructed as a subspace of $\mathbb{R}^3$ as follows: Embed $X$ canonically in the unit square of the xz-plane. For each $x\in X\backslash\{x_0\}$ , let $S_x\subseteq \mathbb{R}^3$ be the unique circle whose diameter is the segment from $x_0$ to $x$ and whose normal vector has y-component $0$ . Now we may identify $Y=\{x_0\}\cup \bigcup_{x\in X\backslash\{x_0\}}S_x$ .

The image of the base-arc $B$ in $X$ turns into a disk $D$ in $Y$ . The images of the points $x_n$ trace out a copy of the earring space in $Y$ . Thus $Y$ may be thought of as a disk with a shrinking sequence of 2-dimensional Cantor fans attached (converging to $y_0$ ). The deformation retraction $X\times [0,1]\to B$ induces a deformation retraction $Y\times [0,1]\to D$ of $Y$ onto the disk. Hence, $Y$ is also contractible but $Y$ is locally path connected at the boundary points of $D$ .

Step 3: Finally, let $\ell_1:[0,1]\to D$ be a loop based at $y_0$ that parameterizes the boundary circle of the disk $D$ . Let $F$ be a Cantor fan and $\ell_2:[0,1]\to F$ parameterize an arc of the fan. Let $E$ be the pushout of $\ell_1$ and $\ell_2$ , i.e. by attaching $F$ to $Y$ along the boundary.

It’s clear that $E$ fails to be locally connected at all of its points. Seeing that $E$ is contractible a little less obvious. Let $Z$ be the union of the disk $D$ and the attached fan $F$ defined as a subspace of the plane as illustrated below. The idea is to define a homotopy equivalence $E\to F$ . This is not a deformation retraction! I describe the details in this tweet thread.

The space Z obtained by attaching a Cantor fan F to the disk D

A view of Edward’s continuum from the top (when embedded naturally in 3-space)

Topological Properties: 2-dimensional, contractible, compact metric space. The key feature of Edward’s continuum is that it is not locally connected at any of its points despite being contractible. It is embeddable in $\mathbb{R}^3$ .

Homotopy Type: This space is contractible. All homotopy and shape invariants of $E$ are trivial.

References

[1] R.D. Edwards, A contractible, nowhere locally connected compactum, Abstracts A.M.S. 20 (1999), 494.

Edward’s Continuum

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