Geodesic
An example of a rim-scattered locally connected continuum that is not rim-countable
Sbornik. Mathematics, Tome 83 (1995) no. 1, pp. 177-187 Cet article a éte moissonné depuis la source Math-Net.Ru

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The result presented in the paper is related to the classification of locally connected continua, in particular, according to their rim properties. 
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S. A. Drozdovskii; V. V. Filippov. An example of a rim-scattered locally connected continuum that is not rim-countable. Sbornik. Mathematics, Tome 83 (1995) no. 1, pp. 177-187. http://geodesic.mathdoc.fr/item/SM_1995_83_1_a8/

[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

[2] Engelking R., Obschaya topologiya, Mir, M., 1986 | MR