Some Remarks on the Algebraic Sum of Ideals and Riesz Subspaces

http://dx.doi.org/10.4153/CMB-2011-151-0
Canad. Math. Bull. 56(2013), 434-441

  Published:2011-08-03
 Printed: Jun 2013

Following ideas used by Drewnowski and Wilansky we prove that if $I$ is an infinite dimensional and infinite codimensional closed ideal in a complete metrizable locally solid Riesz space and $I$ does not contain any order copy of $\mathbb R^{\mathbb N}$ then there exists a closed, separable, discrete Riesz subspace $G$ such that the topology induced on $G$ is Lebesgue, $I \cap G = \{0\}$, and $I + G$ is not closed.