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].

Edward’s Continuum: a contractible continuum that is not locally connected at any of its points.

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)
Another view of Edward’s continuum.

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.