Geodesic
Homeomorphisms of function spaces and topological dimension of domains
Serdica Mathematical Journal, Tome 43 (2017) no. 1, pp. 079-092.

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It is known since 1982, that dim X = dim Y whenever the function spaces Cp(X) and Cp(Y) are linearly homeomorphic. This statement was later extended to uniform homeomorphisms of the spaces Cp(X) and Cp(Y). We obtain, in the case of separable function spaces, a generalization of the first result to another direction. We introduce, for each X, some subspace E(X) ⊂ CpCp(X), which is significantly wider, than the space Lp(X) of all linear continuous functionals on Cp(X). Our generalization includes homeomorphisms h : Cp(X) → Cp(Y), such that the image of Y under the dual mapping h^* of h is contained in E(X) and the image of X under (h^−1 ) ^∗ is contained in E(Y).
Keywords: pointwise convergence topology, topological dimension, 54C35 
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Lazarev, Vadim. Homeomorphisms of function spaces and topological dimension of domains. Serdica Mathematical Journal, Tome 43 (2017) no. 1, pp. 079-092. http://geodesic.mathdoc.fr/item/SMJ2_2017_43_1_a5/