A function $f: X→Y$ between topological spaces $X$ and $Y$ is called a Baire-one function, if there exists a sequence of continuous functions $f_n: X→Y$ such that $\lim_{n \to ∞}{f_n(x)}=f(x)$ for every $x∈X$. We denote the collection of all Baire-one functions between $X$ and $Y$ by $B_1(X,Y)$. It is know that the class $B_1 (\mathbb{R},\mathbb{R})$ is closed under uniform limits. Moreover, Karlova and Mykhaylyuk proved in 2006 that the class $B_1(X,Y)$ is closed under uniform limits if $X$ is a topological space and $Y$ is metrizable path-connected and locally path-connected space. From the other hand, it was shown by Karlova and Mykhaylyuk that there exist a path-connected (but not locally path-connected) subset $Y ⊆ \mathbb{R}^2$ and a sequence of Baire-one functions $f_n: [0;1]→Y$ which tends uniformly to a function $f: [0;1]→Y$ such that $f$ does not belong to the first Baire class.

Therefore, it is actual to study spaces $Y$ for which the class $B_1 (X,Y)$ is closed under uniform limits. The notion of an $R$-space was introduced by Karlova, who proved that if $Y$ is an $R$-space, then $B_1 (X,Y)$  is closed under uniform limits for an arbitrary topological space $X$. Unfortunately, the definition of an $R$-space is rather strong in order to include many curves on the plane. For example, a unit circle is an R-space, but an ellipse is not. Consequently, we need to find a weaker condition on the space $Y$ under which $Y$ remains favorable for the question on uniform limits of Baire-one functions.

We introduce a class of weak $R$-spaces which includes convex subsets of normed spaces and some other curves than a circle on the plane, and prove that the uniform limit $f$ of a sequence of Baire-one functions $f_n: X→Y$ between a topological spaces $X$ and a weak $R$-space $Y$ belongs to the first Baire class.

[1] Lukan M. $R$-spaces and uniform limits of sequences of functions, The 13th International Summer School in Analysis, Topology and Applications (July 29 - August 11, 2018, Vyshnytsya, Chernivtsi Region, Ukraine). Book of Abstracts. P. 24-25.

[2] Karlova O. Baire classification of mappings with values in subsets of finite-dimensional spaces, Nauk. Visn. Cherniv. Univ. Mathematics. 239, (2005), 59-65 (in Ukrainian, English summary).

[3] Karlova O., Mykhaylyuk V. Functions of the first Baire class with values in metrizable spaces, Ukr. Math. J. 58 (4) (2006), 567-571 (in Ukrainian, English summary).

[4] Kuratowski K. Topology. Volume 1, Moscow, 1966 (in Russian).

[5] Engelking R. General topology, Moscow, 1986 (in Russian).