Construct a connected compact set but nowhere locally connected.

Question

Can we construct a connected compact set $X$ in $\mathbb{C}$ ,s.t. it is not locally connected at any point in $X$？

My idea: I know that the topologist's sine curve is connected compact, and also not locally connected at some points, but not all. I cannot find other examples.

Any help will be appreciated!

score 3 · Answer 1 · answered Aug 21 '20 at 01:27

3

The pseudo-arc and the Brouwer–Janiszewski–Knaster continuum (buckethandle) both work.

This kind of question falls within the realm of "continuum theory" which dives deep into these kinds of topological phenomena.

answered Aug 21 '20 at 01:27

Jeremy Brazas

2,247

Another example in the same spirit is the Smale-Williams attractor. Wikipedia has a nice animation of its construction. – Mike F Aug 21 '20 at 01:55
More generally, any indecomposable continuum has the desired properties, so searching on indecomposable continuum is likely to be useful. – Brian M. Scott Aug 21 '20 at 02:34

Construct a connected compact set but nowhere locally connected.

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