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The abstract of the talk (see below) may be downloaded in one of the following formats:
Let $F$ be a vector- lattice of real functions on a set $X$ and let $\cl(F)$ be
its uniform closure. This talk is devoted to the study of conditions on $F$, in
order that $\cl(F)$ has certain algebraic properties.
Specifically, we analyse the problems of knowing when $\cl(F)$ is closed under
composition with all the real uniformly continuous functions over $\R$, when it
is a ring, or it is closed under composition with all the continuous function
over $\R$, or with all the continuous function over the open sets of $\R$.
It will be noticed that, if $F$ contains unbounded functions, each one of these
problems is different to each other. For instance, if $\cl(F)$ is a uniformly
closed ring, then it is also closed under composition with the functions of the
ring generated by the polynomials and the functions of $C(\R)$ that vanish at
infinity, but not under all the functions in $C(\R)$. Nevertheless, most of the
results presented here have been obtained by applying a common technique that
involves certain kind of countable covers of $X$, the so-called 2-finite
covers.
\begin{thebibliography}{10}
\bibitem{GM92}
M.\,I.~Garrido, F.~Montalvo,
{\em Uniform approximation theorems for real-valued
continuous functions},
Topology Appl. {\bf45} (1992), 145--155.
\bibitem{GM96}
M.\,I.~Garrido, F.~Montalvo,
{\em Algebraic properties of the uniform closure of spaces of continuous
functions},
Annals of the New York Academy of Sciences,
{\bf788} (1996), 101--107.
\bibitem{GM99}
M.\,I.~Garrido, F.~.~Montalvo,
{\em Countable covers and uniform closure},
Rend. Ist. Mat. Univ. Trieste, Supp. Vol. XXX (1999), 91--102.
\bibitem{Mrowka68}
S.~Mrowka,
{\em On some approximation theorems},
Nieuw Archieef voor Wiskunde,
XVI (1968), 94--111.
\end{thebibliography}