Geodesic
Components and inductive dimensions of compact spaces
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 2, pp. 445-456.

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It is shown that for every pair of natural numbers $m\geq n\geq 1$, there exists a compact Fréchet space $X_{m,n}$ such that \item {(a)} $\mathop{\rm dim}X_{m,n}=n$, $\mathop{\rm ind}X_{m,n}=\mathop{\rm Ind}X_{m,n}=m$, and \item {(b)} every component of $X_{m,n}$ is homeomorphic to the $n$-dimensional cube $I^n$. \endgraf \noindent This yields new counter-examples to the theorem on dimension-lowering maps in the cases of inductive dimensions.
Classification : 54F45
Keywords: inductive dimension; theorem on dimension-lowering maps; component. 
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     author = {Krzempek, Jerzy},
     title = {Components and inductive dimensions of compact spaces},
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Krzempek, Jerzy. Components and inductive dimensions of compact spaces. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 2, pp. 445-456. http://geodesic.mathdoc.fr/item/CMJ_2010__60_2_a11/