On caractérise, à l’aide de la notion algébrique d’idéal réel, les idéaux fermés de type fini de l’anneau $E\left({\mathbf{R}}^2\right)$ des fonctions différentiables sur ${\mathbf{R}}^2$ ayant la propriété des zéros, et les idéaux fermés principaux de $E\left({\mathbf{R}}^3\right)$ ayant la propriété des zéros.

Linear topological properties of the Lumer-Smirnov class $L{N}_{\ast }{(}^n)$ of the unit polydisc ${}^n$ are studied. The topological dual and the Fréchet envelope are described. It is proved that $L{N}_{\ast }{(}^n)$ has a weak basis but it is nonseparable in its original topology. Moreover, it is shown that the Orlicz-Pettis theorem fails for $L{N}_{\ast }{(}^n)$.

Let X be a compact Hausdorff space and M a metric space. $E_0(X,M)$ is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ X with f constant on some neighborhood of x. We describe some general classes of X for which $E_0(X,M)$ is all of C(X,M). These include βℕ, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of $E_0(X,M)$ as a normed linear space. We also build three first countable Eberlein...