A continuum is a compact, connected, metrizable space.
What are examples of continua that are contractible but nowhere locally connected, meaning that no point has a neighbourhood basis consisting of connected sets?
The following is an example of a contractible nowhere locally connected metrizable space, but I'm not aware of any compact examples (every segment on the "main line" has segments sticking out on a dense set).
The question was reposted and answered on MO. Turns out that there are contractible continua that are nowhere locally connected!