Positivity 7: 33–40, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
On Regular Riesz Subspaces
Faculty of Mathematics and Computer Science, A. Mackiewicz University, Umultowska 87, Pozna´n,
Poland. E-mail: email@example.com
Abstract. The paper is devoted to investigations of properties of regular Riesz subspaces and con-
nections between regularity and some topological properties. The problem if a topological closure
preserves regularity is solved in the class of discrete Riesz spaces. We also characterize Dede-
kind complete Riesz spaces possessing the same classes of σ -regular and regular Riesz subspaces
Moreover, various examples of regular and non regular Riesz spaces are presented.
Mathematics Subject Classiﬁcation 1991: Primary: 46A40.
Key words: regular Riesz subspace, σ -regular Riesz subspace, locally solid Riesz spaces, countable
sup property, Lebesgue property
Throughout the paper E and F are an Archimedean Riesz space and its Riesz
subspace, respectively. We refer the reader to [1,4,9] for the basic terminology and
notation concerning Riesz spaces (=vector lattices) and locally solid (topological)
Riesz spaces (TRS). The least upper bound of a set A ⊂ F (of a net (x
elements from F will be denoted by sup
). The subscript is important
because we will often consider a supremum, of the same set, in different spaces.
Similarly, if a net (x
) increases in F to x, then we will shortly write x
is reserved for the characteristic function of the set A. The unit vectors,
i.e, the functions 1
, n ∈ N, will be denoted by e
Let us recall that a Riesz subspace F ⊂ E issaidtoberegular (σ -regular).
if the natural embedding of F into E preserves arbitrary (countable) suprema and
inﬁma, i.e., for every subset of F (countable subset of F ) whose supremum or
inﬁnum exists in F , then the supremum of inﬁnum of the same subset exists in E
and is equal to that in F (see [l, Deﬁnition 1.7]).
Regularity (and σ -regularity) can be characterized as follows (see  Theorem
F is regular (σ -regular in E) if and only if every net (sequence) in F decreasing
to zero in F decreases to zero in E.
Moreover a Riesz subspace F regular (σ -regular) in E is also regular (σ -regular)
in G for every Riesz subspace G satisfying F ⊂ G ⊂ E, and vice versa: if F
is regular (σ -regular) in G and G is regular (σ -regular) in E,thenF is regular
(σ -regular) in E (regularity and σ -regularity are transitive).
As far as we know regular and σ -regular Riesz subspaces were not systematic-
ally investigated and there do not exist many papers devoted to this topic. But we