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 be the following arrangement of Cantor fans: Let
be the ternary Cantor set and
be a Cantor set in
. Let
be the union of all the line segments from
to the points in
. Define
and set
. We take
to be the basepoint of
.
Let be the base-arc of
. It’s straightforward to see that
deformation retracts onto
. From this we can see that
is contractible and that it is only locally (path) connected at the left-most point
.

Step 2: Now let be the reduced suspension
of
, that is,
Let be the canonical basepoint of
. Actually, the space
can be constructed as a subspace of
as follows: Embed
canonically in the unit square of the xz-plane. For each
, let
be the unique circle whose diameter is the segment from
to
and whose normal vector has y-component
. Now we may identify
.
The image of the base-arc in
turns into a disk
in
. The images of the points
trace out a copy of the earring space in
. Thus
may be thought of as a disk with a shrinking sequence of 2-dimensional Cantor fans attached (converging to
). The deformation retraction
induces a deformation retraction
of
onto the disk. Hence,
is also contractible but
is locally path connected at the boundary points of
.
Step 3: Finally, let be a loop based at
that parameterizes the boundary circle of the disk
. Let
be a Cantor fan and
parameterize an arc of the fan. Let
be the pushout of
and
, i.e. by attaching
to
along the boundary.
It’s clear that fails to be locally connected at all of its points. Seeing that
is contractible a little less obvious. Let
be the union of the disk
and the attached fan
defined as a subspace of the plane as illustrated below. The idea is to define a homotopy equivalence
. This is not a deformation retraction! I describe the details in this tweet thread.

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 .
Homotopy Type: This space is contractible. All homotopy and shape invariants of are trivial.
References
[1] R.D. Edwards, A contractible, nowhere locally connected compactum, Abstracts A.M.S. 20 (1999), 494.