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