Gavriil Paltineanu
The Technical University of Civil Engineering, Bucharest, Romania

A set of functions $F\subset C(X; I=[0,1])$ has property $\mathcal{V}$, if $1-f\in F$ and $f\cdot g\in F, \forall\, f,g\in F$. Von Neumann is the one who drew attention to the collection of functions with property $\mathcal{V}$ in [4]. Moreover, he claims, without proof, a density theorem for such families of functions. A careful analysis of these sets and their properties was made by R. I. Jewett in [3].

In this paper we present a new and more accessible proof for the majority of the results from [3]. We especially mention our proof of Lemma 1 which plays an essential role in the whole paper [3]. Also we mention some new results, such as Propositions 1 and 2, which make the connection between the Uryson sets and the sets which have property $\mathcal{V}$, and Corollary 3, from which Theorem 4.18 of [5] immediately follows. This is joint work with G. Bucur.

REFERENCES
[1]. I. Bucur and G. Paltineanu, Topics in Uniform Approximation of Continuous Functions, Frontiers in Mathematics, Springer International Publishing, 2020.
[2]. L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton, N.J., 1960.
[3]. R. I. Jewett, A variation on the Stone-Weierstrass theorem, Proc. Amer. Math. Soc. 14(1963), 690-693.
[4]. J. von Neumann, Probabilistic logics and the synthesis of reliable organisms from unreliable components, Automata Studies, pp.93-94, Princeton Univ. Press, Princeton, N.J., 1956.
[5]. G. Paltineanu and I. Bucur, Some Density Theorems in the Set of Continuous Functions with Values in the Unit Interval, Mediterr. J. Math. (2017) 14: 44 DOI 10.1007.