The Lindelöf number is $fu$-invariant

Two Tychonoff spaces $X$ and $Y$ are said to be $l$-equivalent ($u$-equivalent) if $C_p(X)$ and $C_p(Y)$ are linearly (uniformly) homeomorphic. N. V. Velichko proved that the Lindelöf property is preserved by the relation of $l$-equivalence. A. Bouziad strengthened this result and proved that the Lindelöf number is preserved by the relation of $l$-equivalence. In this paper the concept of the support different variants of which can be founded in the papers of S.P. Gul'ko and O.G. Okunev is introduced. Using this concept we introduce an equivalence relation on the class of topological spaces. Two Tychonoff spaces $X$ and $Y$ are said to be $fu$-equivalent if there exists an uniform homeomorphism $h: C_p(Y)\to C_p(X)$ such that $\operatorname{supp}^h x$ and $\operatorname{supp}^{h^{-1}}x$ are finite sets for all $x\in X$ and $y\in Y$. This is an intermediate relation between relations of $u$- and $l$-equivalence. In this paper it has been proved that the Lindelöf number is preserved by the relation of $fu$-equivalence.