Geodesic
An example of a rim-countable locally connected continuum that is not path-connected
Matematičeskie zametki, Tome 65 (1999) no. 5, pp. 659-666 
S. A. Drozdovskii. An example of a rim-countable locally connected continuum that is not path-connected. Matematičeskie zametki, Tome 65 (1999) no. 5, pp. 659-666. http://geodesic.mathdoc.fr/item/MZM_1999_65_5_a2/
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     title = {An example of a rim-countable locally connected continuum that is not path-connected},
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     year = {1999},
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[1] Nikiel J., Tuncali H. M., Tymchatyn E. D., “A locally connected rim-countable continuum which is the continuous image of no arc”, Top. Appl., 42 (1991), 83–93 | DOI | MR | Zbl