Geodesic
On the Brouwer Dimension of Compact Spaces
Matematičeskie zametki, Tome 73 (2003) no. 2, pp. 295-304

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A closed set is called a cut between two disjoint sets if any continuum intersecting both these sets intersects the cut. The main result of this paper is that, for any compact space, the dimension defined by induction on the basis of the notion of cut does not exceed the covering dimension. 
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     author = {V. V. Fedorchuk},
     title = {On the {Brouwer} {Dimension} of {Compact} {Spaces}},
     journal = {Matemati\v{c}eskie zametki},
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     publisher = {mathdoc},
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     number = {2},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_73_2_a13/}
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V. V. Fedorchuk. On the Brouwer Dimension of Compact Spaces. Matematičeskie zametki, Tome 73 (2003) no. 2, pp. 295-304. http://geodesic.mathdoc.fr/item/MZM_2003_73_2_a13/