Linear homeomorphisms of topological almost modules of continuous functions and coincidence of dimension

In this paper, the space of continuous functions $C_p(X,G)$, where $G$ is a topological space, is considered. If the set $G$ is endowed with an almost ring structure, the set $C_p(X,G)$ is a topological almost module. It is proved that the dimension $dim$ of the topological space $X$ is an isomorphic invariant of its topological almost module $C_p(X,I)$, where $I=[0,1)$ is a naturally defined almost ring.
This statement is based on ideas of G. G. Pestov’s work «The coincidence of dimension $dim$ of $l$-equivalent topological spaces», where the following theorem was formulated: if $C_p(X,\mathbf{R})$ and $C_p(Y,\mathbf{R})$ are linearly homeomorphic spaces, then $dim\, X = dim\, Y$. Here, $X$ and $Y$ are arbitrary totally regular spaces, and $C_p(X,\mathbf{R})$ is the space of all continuous real functions on $X$ with the pointwise convergence topology. Note that Pestov’s theorem was generalized to the case of uniform homeomorphisms by S. P. Gul'ko.